A group blog on math, physics and philosophy

**David Corfield:**The Two Cultures Challenge Revisited (August 3, 2021)- Categorifying Cauchy-Schwarz
**Tom Leinster:**Diversity and the Mysteries of Counting (July 30, 2021)- Diversity of order 1 (the exponential of Shannon entropy) arises from the magnitude of a category.
**David Corfield:**Topos Theory and Measurability (July 28, 2021)- Applying topos theory to measure theory.
**Tom Leinster:***Entropy and Diversity*Is Out! (July 24, 2021)- My new book is out: on biological diversity, category theory, and much much more besides.
**Tom Leinster:**Borel Determinacy Does Not Require Replacement (July 23, 2021)- Despite what they say.
**Tom Leinster:**Large Sets 13 (July 22, 2021)- The final post: a summary of the whole series.
**Mike Shulman:**Large Sets 12.5 (July 19, 2021)- Comparing replacement/reflection/collection axioms for ETCS
**Tom Leinster:**Large Sets 12 (July 18, 2021)- Replacement is a natural supplement to ETCS.
**John Baez:**Logical and Sublogical Functors (July 14, 2021)- I’m trying to understand sublogical functors, so I’m looking for examples of sublogical functors between presheaf categories, preferably direct images (if that’s possible).
**Tom Leinster:**Large Sets 11 (July 13, 2021)- Measurability, continued.
**Tom Leinster:**Large Sets 10 (July 10, 2021)- Measurability.
**Mike Shulman:**Large Sets 9.5 (July 8, 2021)- On hyper-inaccessible and Mahlo sets
**Tom Leinster:**Large Sets 9 (July 6, 2021)- On inaccessible sets.
**Tom Leinster:**Large Sets 8 (July 2, 2021)- Some cardinal arithmetic.
**Tom Leinster:**Large Sets 7 (June 29, 2021)- Beth fixed points.
**Tom Leinster:**Large Sets 6 (June 24, 2021)- Introducing the beths.
**Tom Leinster:**Large Sets 5 (June 20, 2021)- The alephs from a categorical point of view.
**Simon Willerton:**Optimal Transport and Enriched Categories II (June 19, 2021)- Read about how the dual transport problem arises from general linear programming duality (and how that arises from general Lagrangian duality).
**Tom Leinster:**Large Sets 4 (June 17, 2021)- The index of a set.
**Tom Leinster:**Large Sets 3 (June 13, 2021)- Brief sketch of the role of well-ordered sets in categorical set theory.
**John Baez:**Data Visualization Course (June 11, 2021)- A data visualization bootcamp for students.
**Tom Leinster:**Large Sets 2 (June 10, 2021)- Part 2 of a series on large cardinals in categorical set theory: limits.
**Tom Leinster:**Large Sets 1 (June 8, 2021)- Introductory post in a new series on large cardinals in categorical set theory.
**Simon Willerton:**Optimal Transport and Enriched Categories I (June 6, 2021)- Read about the optimal transport problem and its dual.
**John Baez:**Schur Functors and Categorified Plethysm (June 2, 2021)- John Baez, Joe Moeller and Todd Trimble have a new paper on “Schur functors and categorified plethysm”. Here’s an explanation.
**Emily Riehl:**Structure vs. Observation (May 14, 2021)- Guest post from the Applied Category Theory School 2021.
**Tom Leinster:**The Just Mathematics Collective (April 30, 2021)- A new organization aiming to shift the global mathematics community towards justice.
**John Baez:**Compositional Robotics (April 20, 2021)- There’s a workshop on applying category theory to robotics on May 31st, 2021.
**John Baez:**Applied Category Theory 2021 — Call for Papers (April 16, 2021)- The deadline for submitting your paper to Applied Category Theory 2021 is May 12th.
**Tom Leinster:**Algebraic Closure (April 13, 2021)- A proof of the existence of algebraic closures, after Conrad, after Zorn.
**John Baez:**Can We Understand the Standard Model Using Octonions? (March 31, 2021)- Two talks on the Standard Model of particle physics.
**John Baez:**Native Type Theory (Part 3) (March 29, 2021)- The third episode of “Native Type Theory” by Christian Williams.
**John Baez:**Native Type Theory (Part 2) (March 21, 2021)- Christian Williams on his new paper with Mike Stay: “Native Type Theory”.
**John Baez:**A Group Theory Problem (March 18, 2021)- I’ve got a problem with Lie groups.
**John Baez:**Can We Understand The Standard Model? (March 17, 2021)- A talk “Can we understand the Standard Model?” on Monday April 5th at noon Eastern Time.
**John Baez:**Mathematics in the 21st Century (March 17, 2021)- A talk on “Mathematics in the 21st Century” at the Topos Institute Colloquium on Thursday March 25, 2021 at 18:00 UTC.
**John Baez:**Emerging Researchers in Category Theory (March 14, 2021)- Eugenia Cheng is an expert on giving clear, fun math talks. Now you can take a free class from her on how to give clear, fun math talks! You need to be a grad student in category theory, and…
**Mike Shulman:**Parametric adjoints = multi-adjoints (March 8, 2021)- A functor is a parametric right adjoint almost exactly when it has a left multi-adjoint.
**Emily Riehl:**Postdoctoral Position in Higher Category Theory at Johns Hopkins University (March 2, 2021)- Job announcement for a postdoctoral position in higher category theory, homotopy type theory, especially as related to quantum logic or quantum field theory at Johns Hopkins.
**David Corfield:**Funding of the nLab (March 2, 2021)- Fund the nLab.
**John Baez:**Theoretical Physics in the 21st Century (March 1, 2021)- A talk on “Theoretical physics in the 21st century” on Monday, 8 March 2021, 15:45 UTC.
**John Baez:**Native Type Theory (Part 1) (February 20, 2021)- Christian Williams on his new paper with Mike Stay: “Native Type Theory”.
**John Baez:**Applied Category Theory 2021 (February 17, 2021)- You can now submit papers to the Fourth Annual International Conference on Applied Category Theory (ACT 2021), July 12-16, 2021.
**John Baez:**Isadore Singer, 1924-2021 (February 15, 2021)- Singer died on February 12, 2021.
**John Baez:**The Mess at Leicester (February 12, 2021)- The London Mathematical Society has taken a stand on the proposed layoffs of mathematicians at the University of Leicester.
**David Corfield:**Tangent ∞-Categories and Cohesion (February 2, 2021)- Tangent categories and cohesion.
**John Baez:**Structured vs Decorated Cospans (January 31, 2021)- How is coronavirus related to the symmetric monoidal Grothendieck construction?
**John Baez:**Nishan Canagarajah Screws Up (January 30, 2021)- Please sign a petition to help keep pure mathematics at Leicester.
**John Baez:**Open Systems: A Double Categorical Perspective (Part 3) (January 24, 2021)- The new improved decorated cospans!
**Mike Shulman:**Postdoctoral Position in HoTT at the University of San Diego (January 20, 2021)- Announcing a postdoc position at USD.
**Mike Shulman:**Categories of Nets (Part 2) (January 19, 2021)- A more detailed description of $\Sigma$-nets and how they generate symmetric monoidal categories.
**John Baez:**Categories of Nets (Part 1) (January 17, 2021)- John Baez, Fabrizio Genovese, Jade Master and Mike Shulman wrote a paper together on Petri nets and their variants.
**John Baez:**This Week’s Finds (1–50) (January 12, 2021)- The first 50 issues of This Week’s Finds are now available in LaTeX.
**Emily Riehl:**Postdoctoral Position in HoTT at Johns Hopkins University (January 3, 2021)- Job announcement for a postdoctoral position in homotopy type theory at Johns Hopkins.
**John Baez:**Applied Category Theory 2021 Adjoint School (January 2, 2021)- Do you want to get involved in applied category theory? Are you willing to do a lot of work and learn a lot? Then this is for you: Applied Category Theory 2021 – Adjoint School. Applications due Friday 29…
**John Baez:**Azat Miftakhov (December 29, 2020)- Please sign a petition protesting the torture of mathematics graduate student Azat Miftakhov.
**John Baez:**Octonions and the Standard Model (Part 11) (December 19, 2020)- If you treat the exceptional Jordan algebra as a 27-dimensional spacetime, what does particle physics look like in this spacetime?
**John Baez:**Octonions and the Standard Model (Part 10) (December 16, 2020)- Formally real Jordan algebras, their symmetry groups, their invariant structures — and how they connect quantum mechanics, special relativity and projective geometry.
**Tom Leinster:**The Lie of “It’s Just Math” (December 13, 2020)- Jade Master’s call to action: “Fellow mathematicians, it’s time to stop letting the military benefit from our work.”
**John Baez:**Bernoulli Numbers and the J-homomorphism (December 10, 2020)- How Bernoulli numbers are connected to the stable homotopy groups of spheres — an elementary intro.
**John Baez:**The Algebraic K-Theory of the Integers (December 9, 2020)- A very short intro to the algebraic K-theory of the integers.
**John Baez:**Applied Compositional Thinking for Engineers (December 7, 2020)- Starting January 7, 2021 you can take a course on applied category theory for engineers.
**John Baez:**Mathematical Phantoms (December 6, 2020)- A ‘mathematical phantom’ is a mathematical object that doesn’t exist in a literal sense, but nonetheless acts as if it did.
**David Corfield:**The Liquid Tensor Experiment (December 6, 2020)- A challenge to the formalised mathematics community
**Tom Leinster:***Entropy and Diversity: The Axiomatic Approach*(December 4, 2020)- New book out on the arXiv: “Entropy and Diversity: The Axiomatic Approach”
**Emily Riehl:**Entropy and Diversity on the arXiv (December 4, 2020)- Announcement of Tom Leinster’s “Entropy and Diversity.”
**John Baez:**Octonions and the Standard Model (Part 9) (November 29, 2020)- Duality in projective plane geometry, and how it lets us break the the Lie group E6 into the Lorentz group, left-handed and right-handed spinors, and scalars in 10d Minkowski spacetime.
**David Corfield:**Dependent Type Theory as an n-Theory (November 27, 2020)- Dependent type theory and philosophy
**Tom Leinster:**The Uniform Measure (November 24, 2020)- A canonical measure on more or less any compact metric space.
**John Baez:**The Tenfold Way (November 22, 2020)- I now have a semiannual column in the Notices of the American Mathematical Society! I’m excited: it gives me a chance to write short explanations of cool math topics and get them read by up to 30,000 mathematicians. It’s…
**Simon Willerton:**Stellenbosch is Hiring (November 18, 2020)- Guest post by Bruce Bartlett The sunny campus of Stellenbosch University in South Africa is hiring! We’re looking to make a permanent appointment of a mathematician at Lecturer or Senior Lecturer level. Job description and application webpage. Deadline 30…
**Tom Leinster:**Magnitude Homology of Enriched Categories and Metric Spaces (November 17, 2020)- New, more accessible version of the foundational paper on magnitude homology of metric spaces.
**Tom Leinster:**The Categorical Origins of Lebesgue Integration, Revisited (November 11, 2020)- Simple universal properties uniquely characterizing the Lebesgue L^p spaces.
**John Baez:**Octonions and the Standard Model (Part 8) (November 10, 2020)- A more geometrical outlook on the connection between 10d Minkowski spacetime and the exceptional group E6.
**David Corfield:**Questions about Questions (November 9, 2020)- dependent type-theoretic account of questions
**John Baez:**Octonions and the Standard Model (Part 7) (November 6, 2020)- Greg Egan, John Huerta and I found an explicit description of the Lie group $\mathrm{E}_6$ in terms of 10d spacetime geometry.
**John Baez:**Octonions and the Standard Model (Part 6) (November 1, 2020)- We describe the determinant on the exceptional Jordan algebra in terms of 10-dimensional spacetime geometry.
**John Baez:**Octonions and the Standard Model (Part 5) (October 29, 2020)- We describe vectors and spinors in 10d Minkowski spacetime using octonions, as a warmup for understanding the exceptional Jordan algebra.
**John Baez:**Epidemiological Modeling With Structured Cospans (October 21, 2020)- Structured cospans and open Petri nets are now embodied in working software!
**John Baez:**No New Normed Division Algebra Found! (October 20, 2020)- Joy Christian’s paper has been retracted: there is no associative real normed division of dimension 8.
**John Baez:**New Normed Division Algebra Found! (September 23, 2020)- Extraordinary claims require extraordinary evidence.
**John Baez:**Special Numbers in Category Theory (September 17, 2020)- There are a few theorems in abstract category theory in which specific numbers play an important role.
**John Baez:**Open Systems: A Double Categorical Perspective (Part 2) (September 16, 2020)- The problems with decorated cospans.
**John Baez:**Symmetric Pseudomonoids (September 15, 2020)- A question about symmetric pseudomonoids.
**John Baez:**Riccati Equations and the Projective Line (September 7, 2020)- Riccati equations are connected to the geometry of the projective line.
**John Baez:**Three Phases of Continued Fraction Theory (September 6, 2020)- Three phases of work on continued fractions: the Greeks, Euler and Gauss.
**Emily Riehl:**Announcing the Johns Hopkins (Virtual) Category Theory Seminar (September 3, 2020)- Announcing the Johns Hopkins category theory seminar, which will be held online during the second half of 2020.
**John Baez:**Chasing the Tail of the Gaussian (Part 2) (September 3, 2020)- We can solve a puzzle of Ramanujan by integrating a Gaussian with continued fractions. It’s weird.
**David Corfield:**Making Life Hard For First-Order Logic (September 3, 2020)- Dependency structure in natural language.
**John Baez:**Chasing the Tail of the Gaussian (Part 1) (August 31, 2020)- The story of a continued fraction rediscovered by Ramanujan.
**John Baez:**Sphere Spectrum Analogue of PGL(2,Z) (August 31, 2020)- What’s the analogue of PGL(2,Z) with the sphere spectrum replacing the integers?
**John Baez:**Euler’s Continued Fraction Formula (August 30, 2020)- A continued fraction for pi/4.
**David Corfield:**Formal and Material Inference (August 23, 2020)- Distinguishing formal and material inference
**John Baez:**Open Systems: A Double Categorical Perspective (Part 1) (August 15, 2020)- Kenny Courser’s thesis “Open Systems: A Double Categorical Perspective” is on the arXiv now.
**John Baez:**Pseudocommutative 2-Monads (August 11, 2020)- A little intro to commutative monads, and some questions about their categorified analogues.
**John Baez:**The Group With No Elements (August 10, 2020)- Learn about the empty group.
**John Baez:**Diary 2003–2020 (August 9, 2020)- You can nab a free copy of my diary.
**John Baez:**Open Systems in Classical Mechanics (August 5, 2020)- You can do classical mechanics compositionally, one piece at a time.
**John Baez:**Octonions and the Standard Model (Part 4) (August 3, 2020)- How to find the Standard Model gauge group inside the symmetries of the exceptional Jordan algebra.
**John Baez:**Linear Logic Flavoured Composition of Petri Nets (July 27, 2020)- In this first post from the ACT2020 Adjoint School, Elena Di Lavore and Xiaoyan Li explain Carolyn Brown and Doug Gurr’s paper “A categorical linear framework for Petri nets”.
**John Baez:**Octonions and the Standard Model (Part 3) (July 24, 2020)- We show that the automorphisms of the octonions fixing a square root of -1 form the group SU(3) - and this group acts on octonions just as it does on the color states of a quark and a lepton.
**John Baez:**Octonions and the Standard Model (Part 2) (July 22, 2020)- A prettier formula for octonion multiplication.
**John Baez:**Octonions and the Standard Model (Part 1) (July 17, 2020)- There’s a way to build the octonions from complex scalars and vectors which makes it evident how SU(3) acts as automorphisms fixing some square root of -1 in the octonions.
**John Baez:**Self-Referential Algebraic Structures (July 8, 2020)- In some algebraic structures, each element of the structure acts as an automorphism or derivation of that structure.
**Tom Leinster:**Congratulations, John! (July 2, 2020)- John Baez wins graduate mentoring award
**John Baez:**Getting to the Bottom of Noether’s Theorem (June 29, 2020)- What is Noether’s theorem really trying to tell us about physics?
**John Baez:**ACT2020 Program (June 27, 2020)- Here’s the program for ACT2020, which will take place July 7-10 2020.
**John Baez:**ACT2020 Tutorial Day (June 17, 2020)- Register for the applied category theory tutorials that are taking place July 6, 2020 as part of ACT2020!
**John Baez:**Categorical Statistics Group (June 11, 2020)- Here is a reading list on statistics for category theorists.
**John Baez:**Jordan Algebras (June 7, 2020)- What’s the conceptual meaning of the Jordan algebra axioms?
**Tom Leinster:**Categorical Probability and Statistics 2020 (June 1, 2020)- Workshop this coming weekend on categorical probability and statistics
**Simon Willerton:**Online Magnitude Talk by Mark Meckes (May 30, 2020)- See Mark Meckes give a online talk on June 6th
**John Baez:**The Brauer 3-Group (May 16, 2020)- Why the Brauer 3-group can be described using a chain complex.
**David Corfield:**In Further Praise of Dependent Types (May 10, 2020)- Dependent types in natural language and mathematics.
**John Baez:**Group Cohomology and Homotopy Fixed Points (April 28, 2020)- An explanation of crossed homomorphisms, leading up to a sketch proof of Hilbert’s Theorem 90.
**Mike Shulman:**Model Categories as a Chu Construction (April 28, 2020)- Model categories and Quillen multivariable adjunctions can be embedded in a 2-Chu construction.
**John Baez:**Crossed Homomorphisms (April 24, 2020)- What’s the conceptual meaning of crossed homomorphisms?
**David Corfield:**Ultracategories and 2-Monads (April 16, 2020)- ultracategories
**Tom Leinster:**Online Seminar Lists (April 15, 2020)- A list of lists of online seminars.
**John Baez:**Western Hemisphere Colloquium on Geometry and Physics (April 9, 2020)- This biweekly online colloquium features geometers and physicists presenting current research on a wide range of topics in the interface of the two fields.
**John Baez:**A Categorical View of Conditional Expectation (April 6, 2020)- Prakash Panagaden is giving a talk on category theory and conditional expectations, Wednesday April 8th, 2020.
**John Baez:**Structured Cospans and Petri Nets (April 6, 2020)- A talk on open Petri nets illustrating the use of structured cospans.
**John Baez:**Category Theory Calendar (April 6, 2020)- Here’s a calendar of category theory events, especially online seminars.
**John Baez:**Structured Cospans and Double Categories (March 31, 2020)- The first ACT@UCR seminar is happening online at Wednesday April 1st at 5 pm UTC.
**John Baez:**Online Worldwide Seminar on Logic and Semantics (March 30, 2020)- You can now attend the new Online Worldwide Seminar on Logic and Semantics.
**David Corfield:**Pyknoticity versus Cohesiveness (March 28, 2020)- Pyknotic stuctures
**John Baez:**MIT Categories Seminar (March 25, 2020)- This spring, the MIT Categories Seminar is meeting online each Thursday, 12 noon to 1pm Eastern Time.
**John Baez:**Category Theory Community Server (March 25, 2020)- Christian Williams has started a community server for category theory.
**John Baez:**ACT@UCR Seminar (March 24, 2020)- The applied category theory group at U. C. Riverside is starting an online seminar.
**John Baez:**Applied Category Theory 2020 (Part 2) (March 24, 2020)- The conference Applied Category Theory 2020 will take place July 6-10 — online, worldwide!
**David Corfield:**Michael Harris on Virtues of Priority (March 19, 2020)- Michael Harris’s new paper.
**John Baez:**The Hardest Math Problem (March 14, 2020)- What math problem has taken the most time to be solved? It could be one that’s solved now, or one that’s still unsolved.
**John Baez:**Applied Category Theory 2020 (Part 1) (March 3, 2020)- The big annual conference on applied category theory, ACT2020, is happening July 6–10 at MIT in Cambridge Massachusetts.
**John Baez:**String Diagrams in Computation, Logic, and Physics (March 2, 2020)- The 4th Annual Workshop on String Diagrams in Computation, Logic, and Physics is happening on June 23, 2020 in Bergen, Norway.
**David Corfield:**Type Theory and Propositions (February 26, 2020)- Propositions in HoTT
**John Baez:**What’s the Most Exciting New Mathematics? (February 20, 2020)- What do *you* think are the most exciting new developments in math?
**John Baez:**2-Dimensional Categories (February 17, 2020)- Niles Johnson and Donald Yau have put a 476-page book on _2-Dimensional Categories_ on the arXiv.
**John Baez:**Robert Hermann, 1931–2020 (February 15, 2020)- Robert Hermann died on February 15, 2020.
**David Corfield:**Magidor on Category Mistakes and Context (February 14, 2020)- context-dependent sentences
**David Corfield:**Types in Natural Language (February 11, 2020)- Types in natural language
**David Corfield:**Modal Homotopy Type Theory - The Book (February 6, 2020)- New book - Modal Homotopy Type Theory
**John Baez:**EGA1: The Language of Schemes (February 6, 2020)- An English translation of Grothendieck’s EGA1 is now available, open-source.
**John Baez:**Profunctor Optics: The Categorical View (January 29, 2020)- Emily Pillmore and Mario Román explain profunctor optics in this article written for the ACT2019 Adjoint School.
**Tom Leinster:**Emily Gets A Huge Prize (January 23, 2020)- “Force of nature” Emily Riehl is awarded prestigious research prize.
**John Baez:**Random Permutations (Part 13) (January 17, 2020)- We can embed the groupoid of finite sets with permutation into a slightly larger groupoid with very nice properties.
**Tom Leinster:**Codensity Monads (January 16, 2020)- Whenever you meet a functor, ask “what is its codensity monad?”
**John Baez:**Random Permutations (Part 12) (January 11, 2020)- I’ll describe the groupoid of ‘finite sets equipped with a permutation’ as a sum over Young diagrams. Taking the groupoid cardinality, this description gives a well-known formula for the probability that a random permutation belongs in any given conjugacy class.
**Tom Leinster:**Quotienting Out The Degenerate (January 10, 2020)- Is there a relationship between the definitions of exterior power and normalized chain complex?
**Tom Leinster:**Counting Nilpotents: A Short Paper (December 30, 2019)- New paper: a linear analogue of Joyal’s proof of Cayley’s tree formula, showing how many nilpotent operators there are on a vector space of finite cardinality.
**John Baez:***Compositionality*: First Issue (December 29, 2019)- There’s a new diamond open-access journal on applied category theory, and its first issue has been published!
**Tom Leinster:**Mathematical Images (December 26, 2019)- A collection of (loosely) mathematical images.
**John Baez:**Schrödinger’s Unified Field Theory (December 26, 2019)- In 1947, Schrödinger got really excited about a certain unified field theory. But what did it actually amount to?
**John Baez:**Random Permutations (Part 11) (December 24, 2019)- We can use groupoid cardinality to prove the Cycle Length Lemma, a fundamental result in the theory of random permutations.
**John Baez:**Applied Category Theory 2020 - Adjoint School (December 24, 2019)- The deadline to apply for the next applied category theory school is January 15, 2020.
**Tom Leinster:**A Joyal-Type Proof Of The Number Of Nilpotents (December 20, 2019)- How many nilpotents are there on a finite vector space? A new (?) proof mimicking Joyal’s beautiful proof of the Cayley formula.
**John Baez:**Random Permutations (Part 10) (December 18, 2019)- We can use groupoid cardinality to compute the moments of this random variable: the number of cycles of length k in a random permutation of an n-element set.
**John Baez:**Applied Category Theory Postdocs at NIST (December 13, 2019)- The National Institute of Standards and Technology (NIST) is looking to expand its group of applied category theorists.
**John Baez:**Random Permutations (Part 9) (December 11, 2019)- Treat the number of cycles of length k in a permutation of an n-element set as a random variable. What do the moments of this random variable approach as n approaches infinity?
**John Baez:**Random Permutations (Part 8) (December 8, 2019)- Suppose raindrops are falling on your head, randomly and independently, at an average rate of one per minute. What’s the average of the cube of the number of raindrops that fall on your head in one minute? We can solve this with the help of groupoid cardinalities.
**John Baez:**Random Permutations (Part 7) (December 7, 2019)- As n approaches infinity, the number of cycles of length k in a random permutation of an n-element set approaches a Poisson distribution with mean 1/k.
**Tom Leinster:**A Visual Telling of Joyal’s Proof Of Cayley’s Formula (December 7, 2019)- André Joyal’s beautiful derivation of the number of trees on a given set of vertices
**John Baez:**Position at U. C. Riverside (December 5, 2019)- U. C. Riverside is hiring.
**John Baez:**Topoi from Lawvere Theories (December 5, 2019)- What’s the category of presheaves on a Lawvere theory like?
**David Corfield:**A Couple of Talks in Vienna (December 4, 2019)- Two talks in Vienna
**Tom Leinster:**Counting Nilpotents (December 3, 2019)- What is the probability that a random linear operator is nilpotent?
**John Baez:**Random Permutations (Part 6) (December 1, 2019)- Now we’ll compute the expected number of cycles of length k in a random permutation of an n-element set. This will answer lots of questions about random permutations!
**John Baez:**Random Permutations (Part 5) (November 30, 2019)- Here we’ll use bivariate generating functions to figure out the average number of cycles in a random permutation of an $n$-element set.
**John Baez:**Random Permutations (Part 4) (November 25, 2019)- A random permutation of a large finite set has an almost 70% chance of having one cycle that contains at least half the elements! Let’s see why.
**John Baez:**Random Permutations (Part 3) (November 24, 2019)- An attempt to get a better intuition for a random permutation of a large set.
**Tom Leinster:**Topics in Category Theory: A Spring School (November 24, 2019)- Category theory meeting for early-career researchers in Edinburgh next spring.
**John Baez:**Random Permutations (Part 2) (November 24, 2019)- In the limit as n approaches infinity, what is the probability that the shortest cycle in a randomly chosen permutation of an n-element set has length more than k?
**John Baez:**Random Permutations (Part 1) (November 24, 2019)- What is the probability that a randomly chosen permutation of a large finite set has exactly some chosen number of fixed points?
**John Baez:**The Golomb-Dickman Constant (November 23, 2019)- The Goldman–Dickman constant governs the average length of the longest cycle of a randomly chosen permutation.
**Simon Willerton:**Chair in Pure Mathematics at Sheffield (November 19, 2019)- Apply for a professorship in Sheffield!
**John Baez:**Total Maps of Turing Categories (November 18, 2019)- guest post by Adam Ó Conghaile and Diego Roque We continue the 2019 Applied Category Theory School with a discussion of the paper Total maps of Turing categories by Cockett, Hofstra and Hrubeš. Thank you to Jonathan Gallagher for…
**Tom Leinster:**Geodesic Spheres and Gram Matrices (November 16, 2019)- The geodesic metric on the unit sphere of any inner product space, and why it really is a metric.
**John Baez:**Doubles for Monoidal Categories (November 15, 2019)- As part of the ACT2019 schoo, Fosco Loregian and Bryce Clarke explain Pastro and Street’s paper on doubles for monoidal categories.
**Tom Leinster:**Weak Infinity-Categories Via Terminal Coalgebras (October 30, 2019)- “New” paper with Eugenia Cheng.
**Mike Shulman:**Mixing Internalization with Virtualization: a Terminological Problem (October 29, 2019)- Is there a good terminology for mixing internalization with virtualization?
**John Baez:**Screw Theory (October 21, 2019)- Screw theory is about the Euclidean group. A ‘screw’ is an element of the Lie algebra of this group: an infinitesimal translation together with an infinitesimal rotation.
**Tom Leinster:**Jaynes on Clever Tricks (October 19, 2019)- How important are clever tricks in probability theory?
**John Baez:**Diversity Workshop at UCR (October 14, 2019)- We’re having a workshop on diversity in math on Friday 8 November 2019 at U. C. Riverside, right before the applied category theory meeting that weekend.
**John Baez:**Foundations of Math and Physics One Century After Hilbert (October 10, 2019)- Review of
*Foundations of Mathematics and Physics One Century After Hilbert: New Perspectives* **Tom Leinster:**Edinburgh is Hiring (October 3, 2019)- We’re hiring in areas including category theory.
**John Baez:**Applied Category Theory Meeting at UCR (Part 2) (September 30, 2019)- There will be a meeting on applied category theory on the weekend November 9–10 at U. C. Riverside. Here is the schedule.
**John Baez:**Partial Evaluations 2 (September 16, 2019)- Carmen Constantin from the ACT 2019 Adjoint School talks about “partial evaluations”, weakly cartesian monads, and the bar construction.
**John Baez:**Stellenbosch is Hiring (September 12, 2019)- Stellenbosch University is advertising two permanent faculty positions at the level of Senior Lecturer and Professor.
**John Baez:**The Riemann Hypothesis (Part 3) (September 11, 2019)- How the Weil Conjectures lead us to the idea of ‘motives’.
**John Baez:**The Riemann Hypothesis (Part 2) (September 10, 2019)- A baby version of the Riemann Hypothesis: Hasse’s theorem on elliptic curves.
**John Baez:**The Riemann Hypothesis (Part 1) (September 7, 2019)- What does the Riemann Hypothesis actually mean?
**John Baez:**Homotopy Type Theory Electronic Seminar Talks (September 6, 2019)- The Homotopy Type Theory Electronic Seminar Talks (HoTTEST) will be returning in Fall 2019.
**David Corfield:**The Narratives Category Theorists Tell Themselves (September 3, 2019)- Narratives of a category theorist
**John Baez:**From Simplicial Sets to Categories (August 31, 2019)- I need a reference to a proof of the fact that the left adjoint of the simplicial nerve preserves finite products.
**John Baez:**Turing Categories (August 27, 2019)- Georgios Bakirtzis and Christian Williams discuss Hofstra and Cockett’s paper on Turing categories.
**John Baez:**Graphical Regular Logic (August 16, 2019)- guest post by Sophie Libkind and David Jaz Myers This post continues the series from the Adjoint School of Applied Category Theory 2019….
**John Baez:**Evil Questions About Equalizers (August 16, 2019)- Does the process of converting a category with finite colimits into a symmetric monoidal category give a functor from Rex to SMC that preserves equalizers?
**John Baez:**Even-Dimensional Balls (August 11, 2019)- There’s a simple formula for the sum of the volumes of all even-dimensional balls of radius r. But what does it mean?
**John Baez:**The Conway 2-Groups (August 9, 2019)- There are 24 inequivalent ways to extend the Conway group to a 2-group, i.e. a one-object weak 2-groupoid, where each 1-morphism has a circle’s worth of auto-2-morphisms.
**John Baez:**2020 Category Theory Conferences (August 9, 2019)- Learn the dates and locations for CT2019 and ACT2019.
**David Corfield:**Summer Meanderings About Enriched Logic (July 23, 2019)- Continuous logic as an enriched logic
**John Baez:**Applied Category Theory 2019 Talks (July 22, 2019)- Read about the talks at Applied Category Theory 2019.
**Mike Shulman:**What is the Laplace Transform? (July 17, 2019)- The Laplace Transform is a generalization of Laurent series.
**Tom Leinster:**What Happened At The Magnitude Workshop (July 14, 2019)- Slides from a workshop: Magnitude 2019: Analysis, Category Theory, Applications
**John Baez:**The Riemann Hypothesis Says 5040 is the Last (July 6, 2019)- Riemann’s Hypothesis is equivalent to a simple bound on the sum of divisors of a number.
**David Corfield:**Type Theory, Category Theory and Philosophy (July 5, 2019)- Type theory, Category theory and Philosophy conference
**John Baez:**Structured Cospans (July 1, 2019)- Learn about structured cospans, a new approach to open systems!
**John Baez:**Behavioral Mereology (June 29, 2019)- Toby Smithe and Bruno Gavranovic discuss Fong, Myers, and Spivak’s paper on the concept of “part”.
**John Baez:**Meeting the Dialogue Challenge (June 25, 2019)- Dan Shiebler and Alexis Toumi discuss Grammars as Parsers: Meeting the Dialogue Challenge by Matthew Purver, Ronnie Cann and Ruth Kempson.
**John Baez:**Katrina Honigs meets Grothendieck (June 20, 2019)- A short true story about Grothendieck.
**John Baez:**Applied Category Theory Meeting at UCR (Part 1) (June 16, 2019)- There’s an AMS special session on applied category theory at U.C. Riverside on the weekend of November 9–10, 2019.
**John Baez:**What’s a One-Object Sesquicategory? (June 13, 2019)- What’s a one-object sesquicategory called? A “semi-monoidal category”?
**John Baez:**Nonstandard Models of Arithmetic (June 5, 2019)- Conversations on mathematical logic between Michael Weiss and John Baez, centered on nonstandard models of the Peano axioms of arithmetic.
**John Baez:**Why Category Theory Matters (June 3, 2019)- In which I find that I’m leading a cabal of radical category theorists.
**John Baez:**A Question on Left Adjoints (May 28, 2019)- Jade Master question about the existence of left adjoints for certain functors.
**John Baez:**Polyadic Boolean Algebras (May 23, 2019)- How can we reformulate classical first-order logic more categorically?
**John Baez:**Young Diagrams and Schur Functors (May 20, 2019)- Todd Trimble and I figured out a new way to think about Schur functors, which I explain briefly in these notes.
**John Baez:**Partial Evaluations 1 (May 16, 2019)- A summary of some ideas from the paper “Monads, partial evaluations and rewriting” by Tobias Fritz and Paolo Perrone.
**Mike Shulman:**Right Properness of Left Bousfield Localizations (April 29, 2019)- Right properness of left Bousfield localization corresponds to semi-left-exactness of the localization. (Guest post by Raffael Stenzel.)
**John Baez:**Generalized Petri Nets (April 28, 2019)- Various flavors of Petri net can be handled systematically using the theory of ‘generalized Petri nets’.
**David Corfield:**Twisted Cohomotopy Implies M-Theory Anomaly Cancellation (April 24, 2019)- M-theory and cohomotopy theory.
**John Baez:**Can 1+1 Have More Than Two Points? (April 19, 2019)- Is there a cartesian closed category with finite coproducts such that there exist more than two morphisms from 1 to 1 + 1?
**Mike Shulman:**Univalence in (∞,1)-toposes (April 17, 2019)- A new construction of univalent universes in model categories presenting (∞,1)-toposes
**John Baez:**The ZX-Calculus for Stabilizer Quantum Mechanics (April 14, 2019)- Backens’ completeness proof for the ZX-calculus for stabilizer quantum mechanics.
**John Huerta:**Postdoctoral Researcher Position in Lisbon (April 9, 2019)- You can apply for postdoc position with the “Higher Structures and Applications” research team in Lisbon!
**Tom Leinster:**Category Theory 2019 (April 4, 2019)- Call for registration and talk abstracts
**Simon Willerton:**Motion Group Workshop in Leeds (April 1, 2019)- Read about a meeting in Leeds on loop braid groups.
**John Baez:**Normalising Quantum Circuits (March 24, 2019)- Summary of “A finite presentation of CNOT-dihedral operators”, a paper on quantum circuits by Matthew Amy, Jianxin Chen and Neil J. Ross.
**Simon Willerton:**The Kantorovich Monad (March 24, 2019)- Read Paolo Perrone’s introduction to a probability monad on a category of metric spaces
**John Baez:**Network Models from Petri Nets with Catalysts (March 24, 2019)- We can build a network model from a Petri net with chosen “catalysts”!
**Tom Leinster:**Entropy mod*p*(March 21, 2019)- What is entropy modulo a prime number?
**Mike Shulman:**The Myths of Presentability and the Sharply Large Filter (March 14, 2019)- A pair of plausible-sounding and occasionally-cited claims about presentable objects are in fact false; though in practice it doesn’t matter a whole lot if we are more careful.
**John Baez:**Sporadic SICs and Exceptional Lie Algebras III (March 14, 2019)- There’s a relation between E8 and the problem of fitting as as many equiangular lines as possible into an 8-dimensional
*complex*Hilbert space. **Tom Leinster:**Postdoc at Macquarie (March 11, 2019)- Postdoc available: “Working synthetically in higher categorical structures”.
**Tom Leinster:**How Much Work Can It Be To Add Some Things Up? (March 10, 2019)- Seriously.
**John Baez:**Left Adjoints Between Categories of Posets (March 6, 2019)- A lot of questions about various kinds of posets.
**John Baez:**Sporadic SICs and Exceptional Lie Algebras II (February 28, 2019)- You can find nine equiangular lines in a 3-dimensional complex Hilbert space. They’re related to the Lie algebra E6!
**John Baez:**Tychonoffication (February 20, 2019)- A Tychonoff space is one that’s homeomorphic to a subspace of a possibly infinite-dimensional cube.
**John Baez:**Sporadic SICs and Exceptional Lie Algebras I (February 20, 2019)- Sporadic informationally complete systems of equiangular lines are connected to exceptional Lie algebras!
**John Baez:**Applied Category Theory 2019 (February 7, 2019)- I hope to see you at this conference! Applied Category Theory 2019, July 15-19, 2019, Oxford, UK. Here’s some information about it, such as how to submit papers….
**John Baez:**Jacobi Manifolds (February 4, 2019)- A bit about Jacobi manifolds.
**John Baez:**Symposium on Compositional Structures 3 (January 28, 2019)- SYCO3 is taking place at the University of Oxford on 27-28 March, 2019.
**Simon Willerton:**On the Magnitude Function of Domains in Euclidean Space, IV: Questions and Examples from a Geometric Analyst’s Perspective (b) (January 27, 2019)- See questions in geometry and analysis arising from a geometric analyst’s perspective
**Tom Leinster:**Magnitude Workshop, Edinburgh, July 2019 (January 25, 2019)- Workshop on magnitude in Edinburgh, 4-5 July 2019.
**Simon Willerton:**On the Magnitude Function of Domains in Euclidean Space, III: Questions and Examples from a Geometric Analyst’s Perspective (a) (January 23, 2019)- See questions in topology arising from a geometric analyst’s perspective
**Simon Willerton:**The Passing of Michael Atiyah and Andrew Ranicki (January 21, 2019)- Read about two Edinburgh topologists.
**John Baez:**Applied Category Theory 2019 School (January 5, 2019)- Apply for the Applied Category Theory 2019 school! Applications are due January 30th, 2019.
**John Baez:**HoTT 2019 (December 25, 2018)- There will be an International Conference on Homotopy Type Theory, HoTT 2019, from August 12th to 17th, 2019 at Carnegie Mellon University in Pittsburgh, USA.
**Tom Leinster:**Category Theory 2019 (December 24, 2018)- Invited speakers for Category Theory 2019, and how to submit an abstract.
**John Baez:**Monads and Lawvere Theories (December 23, 2018)- Jade Master has a question about morphisms of Lawvere theories and the corresponding morphisms of monads.
**David Corfield:**Modal Types Revisited (November 18, 2018)- Modal dependent type theory
**Tom Leinster:**Magnitude: A Bibliography (November 15, 2018)- A list of all the papers on magnitude that I know about.
**Tom Leinster:**A Well Ordering Is A Consistent Choice Function (November 12, 2018)- A way of thinking about well orderings.
**Simon Willerton:**More Papers on Magnitude (November 2, 2018)- Look at a list of recent papers on magnitude homology and cohomology.
**John Baez:**2-Groups in Condensed Matter Physics (November 1, 2018)- 2-groups are starting to show up in theoretical condensed matter physics!
**John Baez:**Bar Constructions and Combinatorics of Polyhedra for n-Categories (October 29, 2018)- Samuel Vidal has LaTeXed some notes by Todd Trimble.
**John Baez:**Analysis in Higher Gauge Theory (October 18, 2018)- Andreas Gastel has a new paper ‘Canonical gauges in higher gauge theory’.
**John Baez:**Topoi of G-sets (October 14, 2018)- Reinterpreting basic ideas of group theory in terms of topos theory.
**Tom Leinster:**Category Theory 2019 (October 3, 2018)- Category Theory 2019 will take place at the University of Edinburgh on 7-13 July.
**John Baez:**Applied Category Theory 2019 (October 2, 2018)- Here is a pre-announcement for ACT 2019, an applied category theory conference and school at Oxford, July 15-26, 2019.
**Mike Shulman:**A Communal Proof of an Initiality Theorem (September 26, 2018)- Starting a group project to write out a detailed proof of an initiality theorem for a basic uncomplicated dependent type theory.
**John Baez:**Applied Category Theory Course: Collaborative Design (September 26, 2018)- Here are lectures on enriched profunctors and collaborative design, as part of a course on applied category theory.
**John Baez:**A Pattern That Eventually Fails (September 21, 2018)- Some patterns go on a long time and then fail.
**Mike Shulman:**Cartesian Double Categories (September 20, 2018)- On a paper by Aleiferi about cartesian double categories and their relationship to cartesian bicategories.
**John Baez:***p*-Local Group Theory (September 19, 2018)- What’s the relation between Sylow $p$-subgroups of finite groups, and $p$-completion in algebraic topology?
**John Baez:**What is Applied Category Theory? (September 18, 2018)- Tai-Danae Bradley has a new booklet on applied category theory.
**Tom Leinster:**A Categorical Look at Random Variables (September 5, 2018)- Mark Meckes on random variables (and why probability theory is not about the category of probability spaces)
**John Baez:**Exceptional Quantum Geometry and Particle Physics (August 27, 2018)- Michel Dubois-Violette and Ivan Todorov claim to get the Standard Model gauge group as the subgroup of automorphisms of the exceptional Jordan algebra that 1) preserves a copy of 10d Minkowski spacetime inside this Jordan algebra, and 2) also preserves a unit imaginary octonion—which lets us pick out a copy of 4d Minkowski spacetime inside 10d Minkowski spacetime!
**John Baez:***Compositionality*: Now Open For Submissions (August 24, 2018)- Our new journal is now open for submissions.
**John Baez:**Kan (August 22, 2018)- Jake Bian has created a new add-on for Firefox called ‘Kan’, designed to make the nLab look nicer.
**David Corfield:**The Philosophy and Physics of Noether’s Theorems (August 10, 2018)- Noether anniversary conference
**John Baez:**The Duties of a Mathematician (July 18, 2018)- What are the duties of a mathematician?
**John Baez:***Compositionality*: the Editorial Board (July 18, 2018)- The journal Compositionality has chosen an editorial board.
**John Baez:**Beyond Classical Bayesian Networks (July 7, 2018)- A category-theoretic approach to Bayesian networks, explained by Pablo Andres-Martinez and Sophie Raynor.
**John Baez:**Symposium on Compositional Structures (July 4, 2018)- The First Symposium on Compositional Structures will be taking place at the School of Computer Science of the University of Birmingham, 20-21 September, 2018.
**John Huerta:**Elmendorf’s Theorem (June 27, 2018)- I want to tell you about Elmendorf’s theorem on equivariant homotopy theory. This theorem played a key role in a recent preprint I wrote with Hisham Sati and Urs Schreiber: John Huerta, Hisham Sati, Urs Schreiber - Real ADE-equivariant…
**Mike Shulman:**∞-Atomic Geometric Morphisms (June 15, 2018)- ∞-atomic geometric morphisms appear to coincide with ∞-local-homeomorphisms.
**John Baez:**The Behavioral Approach to Systems Theory (June 15, 2018)- As part of Applied Category 2018 school, Eliana Lorch and Joshua Tan discuss Jan Willems’ behavioral approach to open and interconnected systems.
**John Baez:**Fun for Everyone (June 13, 2018)- Lorscheid’s paper is a great introduction to the mysterious world of ‘the field with one element’.
**John Baez:**Sets of Sets of Sets of Sets of Sets of Sets (June 9, 2018)- There’s no way to make the square of the covariant powerset functor into a monad.
**John Baez:**Applied Category Theory: Resource Theories (June 4, 2018)- An online course on applied category theory now turns to “resource theories”.
**John Baez:**Applied Category Theory: Ordered Sets (May 28, 2018)- Here are 17 lectures on ordered sets, and exercises, and puzzles, and discussions that you can join.
**John Baez:**Laxification (May 25, 2018)- Do you know a systematic way of turning lax monoidal functors into strict ones out of a bigger, “puffed-up” monoidal category?
**John Baez:**Tropical Algebra and Railway Optimization (May 24, 2018)- You can use the tropical rig to help design railway time-tables.
**Emily Riehl:**Postdoc at the Centre of Australian Category Theory (May 20, 2018)- job advertisement at the Centre of Australian Category Theory
**John Baez:**Circuits, Bond Graphs, and Signal-Flow Diagrams (May 19, 2018)- An introduction to Brandon Coya’s thesis on category-theoretic aspects of engineering.
**Mike Shulman:**Linear Logic for Constructive Mathematics (May 19, 2018)- Interpreting linear logic into intuitionistic logic via a Chu construction automatically produces many concepts of traditional constructive mathematics.
**Simon Willerton:**Research Fellowship at the University of Leeds (May 14, 2018)- Apply for a research fellowship in Leeds
**John Baez:***Compositionality*(May 6, 2018)- There’s a new journal,
*Compositionality*, on the study of building big things from smaller things - for example, using category theory. **Tom Leinster:**The Fisher Metric Will Not Be Deformed (May 5, 2018)- One reason why the Fisher metric is so important.
**John Baez:**Props in Network Theory (April 27, 2018)- Brandon Coya and I have a new paper on PROPs and electrical circuits. Here’s an overview.
**Mike Shulman:**What Is an n-Theory? (April 26, 2018)- The models of an (n+1)-theory are the semantic n-theories in that (n+1)-theory, when it is regarded as an “n-doctrine”, i.e. a language in which to write n-theories.
**Simon Willerton:**Torsion: Graph Magnitude Homology Meets Combinatorial Topology (April 11, 2018)- Work through an example of torsion in magnitude homology based on the work of Kaneta and Yoshinaga.
**Simon Willerton:**Magnitude Homology Reading Seminar, II (April 5, 2018)- Read the second installment of the reading seminar notes.
**John Baez:**Dynamical Systems and Their Steady States (April 2, 2018)- At part of the Applied Category Theory 2018 school, Maru Sarazola explains and compares two category-theoretic approaches to open dynamical systems.
**Simon Willerton:**On the Magnitude Function of Domains in Euclidean Space, II (March 25, 2018)- Continue reading about the work of Gimperlein and Goffeng
**Simon Willerton:**Magnitude Homology Reading Seminar, I (March 19, 2018)- See the first in a Sheffield seminar series on the Leinster-Shulman paper on magnitude homology.
**Mike Shulman:**Stabilization of Derivators (March 11, 2018)- A paper on the stabilization of an abstract homotopy theory using derivators as a framework (guest post by Ian Coley).
**John Baez:**Cognition, Convexity, and Category Theory (March 10, 2018)- Tai-Danae Bradley and Brad Theilman, students at the Applied Category Theory 2018 school, continue discussing category-theoretic approaches to linguistics.
**John Baez:**Coarse-Graining Open Markov Processes (March 4, 2018)- Kenny Courser and I have a new paper on coarse-graining open Markov processes, which makes use of symmetric monoidal double categories. Let us know what you think!
**Mike Shulman:**Univalence From Scratch (March 1, 2018)- A self-contained, brief and complete formulation of Voevodsky’s Univalence Axiom.
**John Baez:**Hypergraph Categories of Cospans (February 28, 2018)- As part of the Applied Category Theory school, Jonathan Lorand and Fabrizio Genovese explain Brendan Fong’s work on decorated cospan categories.
**John Baez:**Applied Category Theory at NIST (February 23, 2018)- There will be a workshop on Applied Category Theory at the National Institute of Standards and Technology.
**John Baez:**Higher Algebra and Mathematical Physics (February 23, 2018)- There will be a “double conference” on higher algebra, simultaneously at the Perimeter Institute in Waterloo and the Max Planck Institute in Bonn.
**John Baez:**Cartesian Bicategories (February 19, 2018)- Read Daniel Cicala and Julian Hedges’ summary of the Carboni–Walters paper
*Cartesian Bicategories I*. **David Corfield:**Physics and 2-Groups (February 15, 2018)- Higher gauge theory
**Mike Shulman:**Gradual Typing (February 14, 2018)- A paper modeling gradually typed programming languages as proarrow equipments (guest post by Max New)
**John Baez:**Homotopy Type Theory Electronic Seminar (February 8, 2018)- Watch a series of online talks by the leading experts in homotopy type theory.
**John Baez:**Linguistics Using Category Theory (February 6, 2018)- A summary of the paper “Mathematical Foundations for a Compositional Distributional Model of Meaning” Bob Coecke, Mehrnoosh Sadrzadeh, and Stephen Clark.
**John Baez:***m*Lab (February 5, 2018)- Check out the mLab - a friendly parody of the nLab.
**John Baez:**A Problem on Pushouts and Pullbacks (February 2, 2018)- I’ve got a problem involving a cube in the category of finite sets. I could use some help.
**Tom Leinster:**The Stable Homotopy Hypothesis and Categorified Abelian Groups (January 29, 2018)- Guest post by Nick Gurski
**Emily Riehl:**Announcing the 2018 Talbot Workshop: Model-Independent Theory of Infinity-Categories (January 25, 2018)- Announcing the 2018 Talbot Workshop: Model-Independent Theory of Infinity-Categories
**John Baez:**More Secrets of the Associahedra (January 25, 2018)- How associahedra show up when you invert a formal power series.
**John Baez:**Statebox: A Universal Language of Distributed Systems (January 23, 2018)- Statebox is a startup trying to combine categories, open games, dependent types, Petri nets, string diagrams, and blockchains into a universal language for distributed systems.
**Tom Leinster:**FreeTikZ (January 23, 2018)- Sketch a string diagram by hand, and a new tool by Chris Heunen will automatically convert it into TikZ code.
**John Baez:**A Categorical Semantics for Causal Structure (January 22, 2018)- This is the first blog entry as part of the online reading seminar associated to Applied Category Theory 2018. Joseph Moeller and Dmitry Vagner explain some of Kissinger’s work on causality!
**Simon Willerton:**On the Magnitude Function of Domains in Euclidean Space, I (January 10, 2018)- Read about a recent paper of Gimperlein and Goffeng.
**John Baez:**An M5-Brane Model (December 23, 2017)- The string Lie 2-algebra may be relevant to M5-branes.
**David Corfield:**Arithmetic Gauge Theory (December 21, 2017)- Arithmetic gauge theory
**Mike Shulman:**On Writing Short Papers (December 19, 2017)- How and why to write short papers.
**Simon Willerton:**SageMath and 3D Models in Webpages (December 19, 2017)- Emded 3d models of surfaces into your websites using SageMath.
**Tom Leinster:**Entropy Modulo a Prime (Continued) (December 16, 2017)- The right definition of the entropy of a probability distribution whose “probabilities” are integers mod p.
**Tom Leinster:**Entropy Modulo a Prime (December 14, 2017)- The analogue of Shannon entropy in the field of p elements (after Kontsevich).
**John Baez:**From the Icosahedron to E8 (December 11, 2017)- Here are two ways to get $\mathrm{E}_8$ from the icosahedron. How are they related?
**Mike Shulman:**The 2-Dialectica Construction: A Definition in Search of Examples (December 4, 2017)- Generalized multivariable adjunctions and the Dialectica-Chu construction.
**John Baez:**Real Sets (November 23, 2017)- How to construct a set of cardinality pi.
**Emily Riehl:**Internal Languages of Higher Categories II (November 22, 2017)- Guest post by Chris Kapulkin Two years ago, I wrote a post for the n-Cafe, in which I sketched how to make precise the claim that intensional type theory (and ultimately HoTT) is the internal language of higher category…
**Mike Shulman:**Star-autonomous Categories are Pseudo Frobenius Algebras (November 17, 2017)- Exactly what it says on the tin.
**Mike Shulman:**HoTT at JMM (November 13, 2017)- A special session about homotopy type theory at the 2018 Joint Mathematics Meetings
**John Baez:**Topology Puzzles (November 11, 2017)- Let’s say the closed unit interval [0,1][0,1] maps onto a metric space XX if there is a continuous map from [0,1][0,1] onto XX. Similarly for the Cantor set. Puzzle 0. Does the Cantor set map onto the closed unit…
**Mike Shulman:**The 2-Chu Construction (November 9, 2017)- The 2-categorical Chu construction Chu(Cat,Set).
**Mike Shulman:**The Polycategory of Multivariable Adjunctions (November 7, 2017)- Multivariable adjunctions naturally form a polycategory.
**John Baez:**Applied Category Theory Papers (November 3, 2017)- Some papers on applied category theory.
**Mike Shulman:**Magnitude Homology is Hochschild Homology (November 3, 2017)- A preprint on magnitude homology, with a few refinements and new results.
**John Baez:**The Adjoint School (October 28, 2017)- There’s still some room for people to attend this great school on applied category theory!
**David Corfield:**Categorification and the Cosmic Cube (October 26, 2017)- categorifying cohomology, homological algebra and homotopy
**Tom Leinster:**Unconscious Bias in Recruiting (October 23, 2017)- Online tool for testing whether your job ad is likely to sustain gender inequality.
**John Baez:**Vladimir Voevodsky, June 4, 1966 - September 30, 2017 (October 1, 2017)- Vladimir Voevodsky died on September 30, 2017.
**Simon Willerton:**Lattice Paths and Continued Fractions II (September 22, 2017)- Contemplate further continued fractions.
**John Baez:**Applied Category Theory at UCR (Part 2) (September 22, 2017)- There will be a bunch of talks on applied category theory on November 4th and 5th at U.C. Riverside!
**Simon Willerton:**Lattice Paths and Continued Fractions I (September 18, 2017)- Feel flabbergasted by Flajolet’s Fundamental Lemma
**John Baez:**Applied Category Theory 2018 (September 12, 2017)- Applied Category Theory (ACT 2018) is a conference in Leiden April 30th to May 4th, 2018 preceded by a summer school April 23rd to 27th.
**John Baez:**Postdoc in Applied Category Theory (September 8, 2017)- guest post by Spencer Breiner One Year Postdoc Position at Carnegie Mellon/NIST We are seeking an early-career researcher with a background in category theory, functional programming and/or electrical engineering for a one-year post-doctoral position supported by an Early-concept Grant…
**Simon Willerton:**Schröder Paths and Reverse Bessel Polynomials (August 28, 2017)- See how the reverse Bessel polynomials (which make an appearance in the magnitude of balls) have a combinatorial interpretation.
**John Baez:**A Puzzle on Multi-Sorted Lawvere Theories (August 27, 2017)- I’m curious about when the inclusion into a coproduct is monic.
**John Baez:**Simplicial Sets vs. Simplicial Complexes (August 19, 2017)- What’s the relation between geometrically realizing a simplicial complex directly and doing so indirectly by first turning it into a simplicial set?
**Tom Leinster:**Magnitude Homology in Sapporo (August 11, 2017)- Magnitude homology of metric spaces: the short version.
**John Baez:**A Graphical Calculus for Proarrow Equipments (August 11, 2017)- David Myers has been thinking about string diagrams for calculating with proarrow equipments. Check out the pictures!
**Simon Willerton:**Instantaneous Dimension of Finite Metric Spaces via Magnitude and Spread (August 5, 2017)- Download a talk on instantaneous dimension.
**John Baez:**The Rise and Spread of Algebraic Topology (August 5, 2017)- I’m giving a talk on the history of algebraic topology, and how it led a new conception of mathematics.
**John Baez:**A Compositional Framework for Reaction Networks (July 31, 2017)- Blake Pollard and I have figured out how to describe open chemical reaction networks using the ideas of functorial semantics.
**Emily Riehl:**What is the Comprehension Construction? (July 19, 2017)- This post explains the meaning of “the comprehension construction,” the title of a recent paper by Riehl and Verity (https://arxiv.org/abs/1706.10023).
**Tom Leinster:**Laws of Mathematics “Commendable” (July 14, 2017)- Malcolm Turnbull wants to be able to read your messages.
**John Baez:**A Bicategory of Decorated Cospans (July 8, 2017)- Kenny Courser has a new paper called “A Bicategory of Decorated Cospans”.
**John Baez:**The Geometric McKay Correspondence (Part 2) (July 2, 2017)- Learn how we resolve singularities using the concept of a ‘Hilbert scheme’.
**John Baez:**The Geometric McKay Correspondence (Part 1) (June 19, 2017)- The icosahedron and E
_{8}are connected, with the E_{8}Dynkin diagram arising from a detailed study of how an icosahedron can shrink to zero size. **John Baez:**Eliminating Binders for Easier Operational Semantics (June 10, 2017)- Bound variables cause a lot of complications. In this post Mike Stay explains how to simply avoid them.
**John Baez:**Enriched Lawvere Theories for Operational Semantics (June 7, 2017)- Greg Meredith and Mike Stay are using multisorted Lawvere theories enriched over the category of graphs to do operational sematics!
**Emily Riehl:**A Type Theory for Synthetic ∞-Categories (May 24, 2017)- Summarizes a new paper “A type theory for synthetic $\infty$-categories” by Emily Riehl and Mike Shulman.
**Emily Riehl:**Unboxing Algebraic Theories of Generalised Arities (May 12, 2017)- An introduction to the paper “Monads with arities and their associated theories” written for the Kan Extension Seminar
**Emily Riehl:**A Discussion on Notions of Lawvere Theories (May 1, 2017)- A discussion of the paper “Notions of Lawvere Theory” by Stephen Lack and Jiri Rosicky written for the Kan Extension Seminar
**Tom Leinster:**Functional Equations, Entropy and Diversity: A Seminar Course (April 19, 2017)- Notes from a seminar course.
**Tom Leinster:**The Diversity of a Metacommunity (April 19, 2017)- Conditional entropy, joint entropy and mutual information all have something important to say about the biological diversity of a metacommunity.
**Emily Riehl:**On Clubs and Data-Type Constructors (April 17, 2017)- An introduction to clubs via Kelly’s “On Clubs and Data-Type Constructors” written for the Kan Extension Seminar
**Tom Leinster:**Value (April 14, 2017)- What is the value of the whole in terms of the values of its parts? This apparently very general question, when made precise in a reasonable way, has a surprisingly specific answer.
**John Baez:**Applied Category Theory at UCR (Part 1) (April 5, 2017)- The American Mathematical Society is having a meeting here at U. C. Riverside during the weekend of November 4th and 5th, 2017. I’m organizing a session on Applied Category Theory, and I’m looking for people to give talks.
**Tom Leinster:**Functional Equations IX: Entropy on a Metric Space (April 5, 2017)- A family of entropy measures for probability distributions on metric spaces. Or viewed another way: how to measure biological diversity while taking into account the varying similarities between species.
**Mike Shulman:**Elementary (∞,1)-Topoi (April 4, 2017)- A proposed definition.
**Emily Riehl:**Gluing Together Finite Shapes with Kelly (April 3, 2017)- An exegesis of “Structures defined by limits in the enriched context, I” by Kelly written for the Kan extension seminar
**Emily Riehl:**Enrichment and its Limits (April 3, 2017)- A primer on enriched category theory with a particular focus on weighted limits and colimits, written as a companion for a Kan Extension Seminar post
**John Baez:**Jobs at U. C. Riverside (April 1, 2017)- The Mathematics Department of the University of California at Riverside is trying to hire some visiting assistant professors. We plan to make decisions quite soon! The positions are open to applicants who have PhD or will have a PhD…
**Tom Leinster:**Functional Equations VIII: Measuring Biodiversity (March 29, 2017)- What should the word “diversity” mean for a biological (or other) community, and how is related to the concepts of entropy, mean and norm?
**Tom Leinster:**Functional Equations VII: The*p*-Norms (March 22, 2017)- A theorem saying what’s so special about the p-norms.
**Emily Riehl:**On the Operads of J. P. May (March 21, 2017)- An introduction to Kelly’s paper on the operads of J. P. May
**Tom Leinster:**Functional Equations VI: Using Probability Theory to Solve Functional Equations (March 15, 2017)- How the probabilistic theory of large deviations and the analytic tool of convex conjugacy combine to provide a tool for solving functional equations.
**Mike Shulman:**The Logic of Space (March 10, 2017)- An introduction to type theory, synthetic topology, and homotopy type theory, from a category-theoretic and topological perspective.
**Tom Leinster:**Postdocs in Sydney (March 10, 2017)- Two postdoc positions in category theory, geometry and logic at Macquarie University, Sydney.
**Tom Leinster:**Functional Equations V: Expected Surprise (March 8, 2017)- A family of q-deformations of Shannon entropy, interpreted as expected surprise. Plus, a simple characterization theorem for these “surprise entropies”, and another for relative surprise entropy.
**Emily Riehl:**Algebra Valued Functors in General and Tensor Products in Particular (March 7, 2017)- A summary of Peter Freyd’s article of the same title written for the Kan Extension Seminar.
**Tom Leinster:**Functional Equations IV: A Simple Characterization of Relative Entropy (February 28, 2017)- Relative entropy is almost uniquely characterized by the chain rule.
**Tom Leinster:**Variance in a Metric Space (February 26, 2017)- “Variance” makes sense for probability distributions on a metric space. How should we interpret the variance of the maximum entropy distribution?
**Tom Leinster:**Functional Equations III: Explaining Relative Entropy (February 22, 2017)- An explanation of relative entropy in terms of coding.
**Emily Riehl:**Distributive Laws (February 18, 2017)- An expository account of Beck’s “Distributive Laws” paper written for the Kan Extension Seminar
**Tom Leinster:**Functional Equations II: Shannon Entropy (February 14, 2017)- Shannon entropy and the functional equation that characterizes it.
**David Corfield:**M-theory from the Superpoint (February 13, 2017)- M-theoretic entities as extensions of the superpoint
**Tom Leinster:**The Heilbronn Institute and the University of Bristol (February 10, 2017)- News report on Bristol University and the surveillance state.
**Tom Leinster:**Functional Equations I: Cauchy’s Equation (February 7, 2017)- First instalment of a course on functional equations. This time: which functions f: R –> R satisfy f(x + y) = f(x) + f(y)?
**Emily Riehl:**The Category Theoretic Understanding of Universal Algebra (February 7, 2017)- An expository summary of Hyland and Power’s “The Category Theoretic Understanding of Universal Algebra: Lawvere Theories and Monads” for the Kan Extension Seminar II.
**Tom Leinster:**The Brunn-Minkowski Inequality (January 31, 2017)- A highly general inequality for volume, and how it implies the isoperimetric inequality.
**Tom Leinster:**Papers Written While Drunk (January 24, 2017)- Chemicals in the bloodstream and their influence on mathematics.
**Tom Leinster:**Category Theory in Barcelona (January 10, 2017)- Joachim Kock and I are teaching category theory in Barcelona. We like slogans.
**John Baez:**Globular for Higher-Dimensional Knottings (Part 3) (January 4, 2017)- See a 2-sphere embedded in 4-space!
**Tom Leinster:***Basic Category Theory*Free Online (January 2, 2017)- My introductory textbook Basic Category Theory is now available free online.
**Tom Leinster:**NSA Axes Math Grants (December 31, 2016)- NSA suspends its Mathematical Sciences Program. Fewer moral dilemmas for mathematicians.
**John Baez:**Globular for Higher-Dimensional Knottings (Part 2) (December 17, 2016)- Higher-dimensional surfaces using
*Globular*. **John Baez:**Globular for Higher-Dimensional Knottings (December 16, 2016)- guest post by Scott Carter About 7 months ago, Jamie Vicary contacted me with a Globular worksheet of which, initially, I could make neither heads nor tails. He patiently explained to me that what I was looking at was…
**Tom Leinster:**Field Notes on the Behaviour of a Large Assemblage of Ecologists (December 15, 2016)- What this mathematician made of the British Ecological Society’s annual meeting.
**Tom Leinster:**Quarter-Turns (November 29, 2016)- For which linear operators T is T(x) always orthogonal to x? The answer depends very much on the field you’re working over.
**John Baez:**Linear Algebraic Groups (Part 8) (November 29, 2016)- Learn about algebraic sets and affine algebras!
**Emily Riehl:**Category Theory in Context (November 16, 2016)- Announcing the publication of “Category Theory in Context” by Emily Riehl in Dover’s Aurora: Modern Math Originals series
**John Baez:**Linear Algebraic Groups (Part 7) (November 10, 2016)- Parabolic subgroups, flag varieties, and q-factorials!
**John Baez:**Linear Algebraic Groups (Part 6) (November 7, 2016)- The $q$-deformed version of Pascal’s triangle, and the decomposition of a Grassmannian into Bruhat cells.
**John Baez:**Linear Algebraic Groups (Part 5) (November 3, 2016)- Grassmannians, maximal parabolic subgroups, and a bit of counting!
**John Baez:**Linear Algebraic Groups (Part 4) (October 30, 2016)- Abstract projective planes, Pappus’s hexagon theorem and Klein’s Erlangen program.
**John Baez:***Higher Structures*Journal (October 26, 2016)- There’s a new journal called “Higher Structures”.
**John Baez:**Linear Algebraic Groups (Part 3) (October 26, 2016)- Grassmannians, their Bruhat cells, and how to count their points.
**Emily Riehl:**The Kan Extension Seminar Returns (October 25, 2016)- Announcing a second iteration of the Kan extension seminar, a graduate reading course in category theory.
**John Baez:**Linear Algebraic Groups (Part 2) (October 20, 2016)- Lectures on algebraic groups: the idea of projective geometry.
**John Baez:**Linear Algebraic Groups (Part 1) (October 17, 2016)- Lecture notes on linear algebraic group theory, starting with Euclid.
**Tom Leinster:**Jobs at Edinburgh (October 10, 2016)- Advertisement for two jobs in Algebra, Geometry and Topology at the University of Edinburgh.
**Emily Riehl:**Mathematics Research Community in HoTT (October 4, 2016)- A call for applications to an AMS Mathematics Research Community workshop in HoTT to be held in June 2017.
**John Baez:**Euclidean, Hyperbolic and Elliptic Geometry (September 26, 2016)- What’s the best way to see Euclidean, hyperbolic and elliptic geometry as part of a 1-parameter family.
**Qiaochu Yuan:**Logical Uncertainty and Logical Induction (September 19, 2016)- Progress on the problem of handling logical uncertainty: how do we formally describe our uncertainty about mathematical facts such as the digits of pi or the truth of the Riemann hypothesis?
**Tom Leinster:**Disaster at Leicester (September 15, 2016)- Leicester University is imposing very severe cuts on the Mathematics department. You can add your voice to those objecting.
**David Corfield:**HoTT and Philosophy (September 13, 2016)- HoTT and philosophy
**John Baez:**Twitter (September 12, 2016)- I’m now on Twitter. Watch me tweet.
**Simon Willerton:**Barceló and Carbery on the Magnitude of Odd Balls (September 9, 2016)- Read about Barcel'o and Carbery’s calculation of the magnitude of odd dimensional balls, utilizing the potential theory developed by Meckes.
**John Baez:**The Ultimate Question, and its Answer (September 9, 2016)- What is the Ultimate Question, and what is its Answer?
**Tom Leinster:**Magnitude Homology (September 6, 2016)- A new homology theory for enriched categories, created by Mike Shulman, which looks very much as if it categorifies magnitude.
**Tom Leinster:**Economy of Style (September 3, 2016)- An extreme example of mathematical writing style.
**John Baez:**Topological Crystals (Part 4) (August 28, 2016)- Finally, a gallery of examples of topological crystals!
**Tom Leinster:**Jobs at Heriot-Watt (August 27, 2016)- Permanent positions are available at Heriot-Watt University, Edinburgh.
**Tom Leinster:**Monoidal Categories with Projections (August 25, 2016)- A defence of the technical-sounding concept of “semicartesian monoidal category”.
**John Baez:**Compact Closed Bicategories (August 19, 2016)- Mike Stay’s paper on compact closed symmetric monoidal bicategories is finally published!
**Tom Leinster:**A Survey of Magnitude (August 11, 2016)- Magnitude is a highly informative invariant of metric spaces. Here are some of the highlights of the theory, now collected together in a survey paper by Mark Meckes and me.
**John Baez:**Two Miracles of Algebraic Geometry (August 10, 2016)- Taking the Albanese variety defines an idempotent monad.
**Tom Leinster:**In Praise of the Gershgorin Disc Theorem (August 9, 2016)- An elementary and very useful lemma in basic linear algebra. (Why wasn’t I taught it?)
**Mike Shulman:**What is a Formal Proof? (August 8, 2016)- Is decidable type-checking necessary for a notion of formal proof?
**John Baez:**Topological Crystals (Part 3) (August 6, 2016)- How to build a topological crystal.
**John Baez:**Topological Crystals (Part 2) (July 27, 2016)- Learn about the maximal abelian cover of a graph.
**John Baez:**Topological Crystals (Part 1) (July 23, 2016)- There’s a way to get a “topological crystal” from a graph. Learn how it works!
**Tom Leinster:**How the Simplex is a Vector Space (June 11, 2016)- The interior the n-simplex can be given the structure of a real vector space in a natural way. How, and what use is it?
**John Baez:**Good News (May 26, 2016)- Good news about four former students.
**Mike Shulman:**Castles in the Air (May 20, 2016)- Some thoughts on the value of nonrigorous mathematics.
**David Corfield:**The HoTT Effect (May 19, 2016)- HoTT conference season
**John Huerta:**E_{8}as the Symmetries of a PDE (May 12, 2016)- Work by Dennis The that expresses E
_{8}as the symmetries of an explicit PDE. **John Baez:**The Works of Charles Ehresmann (May 10, 2016)- Charles Ehresmann’s complete works, over 3000 pages of mathematics, are now available for free online.
**Tom Leinster:**Man Ejected from Flight for Solving Differential Equation (May 8, 2016)- Yes, really.
**John Baez:**Which Paths Are Orthogonal to All Cycles? (May 5, 2016)- A question about the first homology group of a graph.
**John Baez:**Categorifying Lucas’ Equation (May 4, 2016)- Gavin Wraith asks: can you find a nice bijective proof that $1^2 + \cdots + 24^2 = 70^2$?
**Qiaochu Yuan:**Relative Endomorphisms (April 30, 2016)- There’s a very general notion of endomorphism object that specializes to endomorphism monads, endomorphism operads, and the codensity monad. Has anyone seen it?
**John Baez:**Polygonal Decompositions of Surfaces (April 23, 2016)- What are the most standard references for polygonal decompositions of compact smooth surfaces, so I can just cite those and use their terminology?
**David Corfield:**Type Theory and Philosophy at Kent (April 21, 2016)- workshop on type theory
**John Baez:**Foundations of Mathematics (March 31, 2016)- Different people have very different attitudes to “foundations”, and until we discuss those, we’ll be talking past each other.
**John Baez:**E_{8}Is the Best (March 24, 2016)- We finally know that E
_{8}gives the densest lattice packing of spheres! **John Baez:**The Involute of a Cubical Parabola (March 23, 2016)- Help me find the icosahedron hidden in the theory of wave propagation in 2 dimensions.
**John Baez:**Prime Numbers and the Riemann Hypothesis (March 21, 2016)- A free book on the connection between prime numbers and the Riemann zeta function!
**Qiaochu Yuan:**Coalgebraic Geometry (March 21, 2016)- Two blog posts on coalgebras in algebraic geometry.
**John Baez:**The Most Common Prime Gaps (March 19, 2016)- What are the most common gaps between primes?
**John Baez:**Weirdness in the Primes (March 15, 2016)- A new discovery shakes up our idea of how random primes are.
**John Baez:**Category Theory Course Notes (March 10, 2016)- Here are the notes for my Winter 2016 introductory course on category theory.
**John Baez:**Category Theory Seminar Notes (March 9, 2016)- Here are the lecture notes for my Fall 2015 introduction to how category theory helps unify our view of mathematics.
**John Baez:**Hyperbolic Kac–Moody Groups (March 4, 2016)- Is there a geometric construction of hyperbolic Kac–Moody groups?
**John Baez:**Integral Octonions (Part 12) (January 31, 2016)- Tim Silverman explores the $\mathrm{E}_8$ lattice mod 2, and its connection with questions we’d studied earlier in this series.
**John Baez:**Thinking about Grothendieck (January 17, 2016)- Barry Mazur has a new short essay on Grothendieck.
**John Baez:**Homotopy of Operads and Grothendieck-Teichmüller Groups (January 16, 2016)- Benoit Fresse has finished a big two-value book on operads, which you can now see on his website.
**John Baez:**A Compositional Framework for Markov Processes (January 12, 2016)- Open Markov processes — that is, Markov processes with specified ‘input’ and ‘output’ states — can be seen as morphisms in a category. This lets us take a big complicated Markov process and think of it as made of smaller open ones.
**Mike Shulman:**On Digital Mathematics and Drive-By Contributors (January 10, 2016)- Slides from a talk about my experiences with digital mathematics.
**John Baez:**Decorated Cospans (January 6, 2016)- Brendan Fong has a paper on ‘decorated cospan categories’. They seem to play a big role in network theory!
**John Baez:**A Compositional Framework for Passive Linear Networks (December 28, 2015)- Yes, you can find categories everywhere — even in electrical engineering! The reason is that we live in a world of networks, and networks tend to be morphisms in categories.
**John Baez:**Operads and Phylogenetic Trees (December 22, 2015)- What do phylogenetic trees have to do with operads? A lot!
**Emily Riehl:**Research Fellowships at Macquarie University (December 8, 2015)- A little bird tells me that Macquarie University is hiring (even more) category theorists! Specifically, they are offering two-year research fellowship positions, details of which can be found here. Macquarie University, which is in greater Sydney, is the home…
**John Baez:**Globular (December 5, 2015)- Jamie Vicary writes about some new software for calculating in $n$-categories.
**John Baez:**42 (December 3, 2015)- In
*The Hitchhiker’s Guide to the Galaxy*, the number 42 was revealed to be the Answer to the Ultimate Question of Life, the Universe, and Everything. But we never learned what the question was! **John Baez:**Burritos for Category Theorists (November 11, 2015)- At last, burritos have been explained in a clear, conceptual way.
**Tom Leinster:**Weil, Venting (November 10, 2015)- Passive-aggressive Weil.
**John Baez:**Cakes, Custard, Categories and Colbert (November 3, 2015)- Eugenia Cheng will be on TV this Wednesday, November 4th.
**David Corfield:**When Not To Use ‘The’ (October 21, 2015)- Conditions for use ‘the’
**John Baez:**Four Tribes of Mathematicians (October 20, 2015)- Read about different motivations for doing mathematics.
**John Baez:**Configurations of Lines and Models of Lie Algebras (Part 2) (October 5, 2015)- Triality, its action on the 24-cell, and the ‘tetrality’ description of $\mathfrak{so}(8)$.
**John Baez:**Configurations of Lines and Models of Lie Algebras (Part 1) (October 4, 2015)- Learn a bit about the 27 lines on a smooth cubic surface.
**Emily Riehl:**An Exact Square from a Reedy Category (September 30, 2015)- Any Reedy category has a canonical square of inclusions of subcategories and this square is exact.
**John Baez:**The Free Modular Lattice on 3 Generators (September 19, 2015)- How is the free modular lattice on 3 generators related to 8-dimensional space?
**Tom Leinster:**Where Does The Spectrum Come From? (September 14, 2015)- A categorical account of the most important invariant of a linear operator.
**Tom Leinster:**Rainer Vogt (September 3, 2015)- An appreciation of Rainer Vogt, who died last month.
**Tom Leinster:**How Do You Handle Your Email? (September 2, 2015)- What’s the best way for academics to manage the day-to-day deluge of email?
**Emily Riehl:**Wrangling Generators for Subobjects (August 31, 2015)- Describes a new categorical definition of the degree of generation of a subobject.
**John Baez:**A Wrinkle in the Mathematical Universe (August 17, 2015)- The outer automorphism of $S_6$ is an exceptional thing.
**John Baez:**Two Cryptomorphic Puzzles (August 9, 2015)- Two puzzles whose solutions are related.
**Mike Shulman:**Internal Languages of Higher Categories (July 28, 2015)- A summary of what we know and don’t know about intensional type theory (HoTT) as an internal language for higher categories (guest post by Chris Kapulkin).
**John Baez:**Category Theory 2015 (July 19, 2015)- You can see slides from Category Theory 2015.
**Tom Leinster:**Mary Shelley on Invention (July 8, 2015)- A thought on invention from the introduction to Frankenstein.
**Mike Shulman:**What is a Reedy Category? (June 29, 2015)- A new preprint explains Reedy categories from a category-theoretic point of view, as certain iterated collages of profunctor diagrams.
**Simon Willerton:**Feynman’s Fabulous Formula (June 29, 2015)- Bruce describes a graph-theoretic result
**Mike Shulman:**What’s so HoTT about Formalization? (June 21, 2015)- What is the relationship between homotopy type theory and computer formalization of mathematics?
**David Corfield:**Carnap and the Invariance of Logical Truth (June 12, 2015)- Awodey on invariance and logic
**John Baez:**Semigroup Puzzles (June 9, 2015)- Two puzzles that involve nothing but playing with equations.
**John Baez:**A 2-Categorical Approach to the Pi Calculus (May 26, 2015)- Greg Meredith and Mike Stay have a new paper on modeling the pi calculus with 2-categories.
**David Corfield:**SoTFoM III and The Hyperuniverse Programme (May 26, 2015)- sotfom iii conference
**John Baez:**PROPs for Linear Systems (May 22, 2015)- PROPs were developed in topology, along with operads, to describe spaces with lots of operations on them. But now some of us are using them to think about ‘signal-flow diagrams’ in control theory—an important branch of engineering. I talked…
**Tom Leinster:**How to Acknowledge Your Funder (May 22, 2015)- Acknowledging the NSA’s support when you object to universal surveillance.
**John Baez:**The Origin of the Word “Quandle” (May 21, 2015)- How did ‘quandles’ get their name?
**Mike Shulman:**The Revolution Will Not Be Formalized (May 18, 2015)- My opinion about the future of computer-verified proof.
**Simon Willerton:**Categorifying the Magnitude of a Graph (May 13, 2015)- See how there’s a homology theory for graphs with magnitude as its Euler characteristic.
**Emily Riehl:**Breakfast at the n-Category Café (April 30, 2015)- Join Michael Harris in discussing homotopy type theory.
**John Baez:**Categories in Control (April 28, 2015)- To understand ecosystems, ultimately will be to understand networks. - B. C. Patten and M. Witkamp A while back I decided one way to apply my math skills to help save the planet was to start pushing toward green…
**Mike Shulman:**A Synthetic Approach to Higher Equalities (April 24, 2015)- An introduction to homotopy type theory for philosophers
**David Corfield:**The Structure of A (April 12, 2015)- How to define the structure of
**John Baez:**Information and Entropy in Biological Systems (April 7, 2015)- You can join a workshop on entropy and information in biological systems, virtually, by watching the streaming video.
**Tom Leinster:**Five Quickies (April 6, 2015)- 1. Reflexive completion. 2. Mathematicians accepting and declining overtures from GCHQ. 3. The Euler characteristic of an algebra. 4. Review of Nick Gurski’s higher categories book. 5. Are lectures the best way to teach?
**John Baez:**Split Octonions and the Rolling Ball (March 14, 2015)- Watch a rolling ball illustrate an exceptional incidence geometry!
**Simon Willerton:**A Scale-Dependent Notion of Dimension for Metric Spaces (Part 1) (March 11, 2015)- Try to understand how dimension can depend on scale
**Mike Shulman:**HITs and the Erlangen Program (March 9, 2015)- Bishop sets and Klein geometries are both quotient constructions, and can both be realized by HITs, in contrast to the univalence/extensionality based approach common in set theory.
**David Corfield:**Mathematics Without Apologies (March 5, 2015)- Book release of ‘Mathematics Without Apologies’
**John Baez:**Concepts of Sameness (Part 4) (February 27, 2015)- Three approaches to defining equality.
**Mike Shulman:**Introduction to Synthetic Mathematics (part 1) (February 26, 2015)- A draft of an introduction to synthetic mathematics
**John Baez:**Concepts of Sameness (Part 3) (February 25, 2015)- What good is the equation x = x?
**John Baez:**Concepts of Sameness (Part 2) (February 23, 2015)- Some notes on the Gongsun Long’s ‘white horse paradox’.
**John Baez:**Concepts of Sameness (Part 1) (February 23, 2015)- Here I’ll start drafting a paper on ‘concepts of sameness’, which will be a chapter in Elaine Landry’s
*Categories for the Working Philosopher*. **John Baez:**Quantum Physics and Logic at Oxford (February 18, 2015)- The 12th International Workshop on Quantum Physics and Logic is being held in Oxford from July 13th to 17th, 2015.
**John Baez:**Can a Computer Solve Lebesgue’s Universal Covering Problem? (February 12, 2015)- Let’s try to prove that the solution to Lebesgue’s universal covering problem is computable.
**John Baez:**Higher-Dimensional Rewriting in Warsaw (February 9, 2015)- There will be a conference on Higher-Dimensional Rewriting and Applications, 28-29 June 2015, in Warsaw.
**Tom Leinster:**More on the AMS and NSA (February 4, 2015)- Letter from cryptographers and mathematicians to the AMS.
**John Baez:**Lebesgue’s Universal Covering Problem (February 4, 2015)- Read about some progress on Lebesgue’s universal covering problem.
**Mike Shulman:**The Univalent Perspective on Classifying Spaces (January 19, 2015)- Defining classifying spaces as full sub-groupoids of a universe can give an easier way to define morphisms between them.
**Tom Leinster:**The AMS Must Justify Its Support of the NSA (January 9, 2015)- Letter in the Notices of the AMS.
**Simon Willerton:**Mathematics and Magic: the de Bruijn Card Trick (January 5, 2015)- Perform a magic trick using the power of maths.
**John Baez:**Braided 2-Groups from Lattices (January 1, 2015)- There are some easy ways to get braided 2-groups from lattices.
**John Baez:**Can One Explain Schemes to a Biologist? (December 31, 2014)- Mumford’s obituary of Grothendieck was rejected by
*Nature*. **Emily Riehl:**A Call for Examples (December 28, 2014)- A call for undergraduate-level examples illustrating categorical ideas
**David Corfield:**Competing Foundations? (December 22, 2014)- foundations conference
**Tom Leinster:**Effective Sample Size (December 18, 2014)- Yet another place where the concept of magnitude turns up: the statistical notion of effective sample size.
**Tom Leinster:**Welcome, Qiaochu! (December 18, 2014)- Qiaochu Yuan joins us as a host of the Café.
**Tom Leinster:**Turing’s Legacy (December 16, 2014)- The legacy of Alan Turing’s work at Bletchley Park (later GCHQ).
**John Baez:**Integral Octonions (Part 11) (December 11, 2014)- Why there are 270 rescaled copies of the first shell of the E
_{8}lattice sitting in the second shell. **Emily Riehl:**A Categorical Understanding of the Proof of Cantor-Schröder-Bernstein? (December 7, 2014)- In search of a categorical interpretation of the proof of the Cantor–Schroeder–Bernstein theorem.
**Tom Leinster:**Graph Colouring and Cartesian Closed Categories (December 5, 2014)- A categorical reformulation of Hedetniemi’s conjecture on colouring a product of graphs.
**John Baez:**Integral Octonions (Part 10) (December 1, 2014)- How is the Leech lattice related to the exceptional Jordan algebra?
**Tom Leinster:**Grothendieck’s Activism And What It Says About The World (November 26, 2014)- Ivar Ekeland’s reflections on the activist phase of Grothendieck’s life.
**John Baez:**Integral Octonions (Part 9) (November 23, 2014)- Learn about using integral octonions and the E
_{8}lattice to construct the Leech lattice. **Tom Leinster:**Barbie on Monads (November 22, 2014)- The definition of monad in a Barbie cartoon.
**Tom Leinster:**General Topology Notes (November 19, 2014)- Undergraduate notes from a course on General Topology.
**Tom Leinster:**The Kan Extension Seminar in the*Notices*(November 18, 2014)- Emily Riehl on running an online graduate reading course.
**Tom Leinster:**Jaynes on Mathematical Courtesy (November 16, 2014)- Edwin Jaynes rails against modern mathematical writing.
**John Baez:**Integral Octonions (Part 8) (November 15, 2014)- With a suitable Lorentzian metric, the lattice of self-adjoint 3 x 3 octonionic matrices with
*integral*octonions as entries becomes isometric to a 27-dimensional lattice that may play a role in bosonic string theory. **John Baez:**Integral Octonions (Part 7) (November 5, 2014)- Greg Egan just showed that $\mathrm{E}_{10}$: is the lattice of $2 \times 2$ self-adjoint matrices with integral octonions as entries! Here’s the proof.
**David Corfield:**Maths, Just in Short Words (October 30, 2014)- Mathematical proofs in short words.
**Tom Leinster:**Where Do Probability Measures Come From? (October 22, 2014)- Tom Avery explains a categorical construction of the concept of probability measure.
**Tom Leinster:**New Evidence of the NSA Deliberately Weakening Encryption (October 17, 2014)- Yet more evidence that the NSA wants to make it impossible to communicate privately.
**David Corfield:**‘Competing Foundations?’ Conference (October 17, 2014)- FINAL CFP and EXTENDED DEADLINE: SoTFoM II `Competing Foundations?’, 12-13 January 2015, London. The focus of this conference is on different approaches to the foundations of mathematics. The interaction between set-theoretic and category-theoretic foundations has had significant philosophical impact, and…
**John Baez:**M-theory, Octonions and Tricategories (October 5, 2014)- John Huerta has completed the fourth of a series of papers on supersymmetry and division algebras.
**Tom Leinster:**The Atoms of the Module World (October 5, 2014)- The one-to-one correspondence between simple and projective indecomposable modules.
**John Baez:**A Packing Pessimization Problem (September 21, 2014)- What convex centrally symmetric shape is worst at tiling the plane with high density?
**Simon Willerton:**Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams (September 10, 2014)- Guest post by Bruce Bartlett I recently put an article on the arXiv: Bruce Bartlett, Quasistrict symmetric monoidal 2-categories via wire diagrams. It’s about Chris Schommer-Pries’s recent strictification result from his updated thesis, that every symmetric monoidal bicategory is…
**David Corfield:**Ronnie Brown in Paris (September 6, 2014)- Ronnie Brown’s recent talk
**John Baez:**Uncountably Categorical Theories (August 31, 2014)- Boris Zilber’s thoughts on ‘logically perfect’ structures.
**Tom Leinster:**Why It Matters (August 25, 2014)- Many mathematicians are uncomfortable about the NSA/GCHQ’s surveillance of entire populations, but fundamentally aren’t that bothered. Does it really matter?
**Tom Leinster:**Holy Crap, Do You Know What A Compact Ring Is? (August 20, 2014)- Every compact ring is totally disconnected.
**John Baez:**The Ten-Fold Way (Part 4) (August 12, 2014)- Todd Trimble’s note on ‘the super Brauer group and super division algebras’, reprinted.
**John Baez:**The Ten-Fold Way (Part 3) (August 7, 2014)- A detailed introduction to the Brauer–Wall monoid, which gives an algebraic explanation of the ten-fold way.
**Tom Leinster:**Wrestling with Tight Spans (August 2, 2014)- A geometric construction, used in combinatorial optimization and phylogenetic analysis, interpreted categorically.
**Tom Leinster:***Basic Category Theory*(July 27, 2014)- My introduction to category theory has just been published.
**John Baez:**The Ten-Fold Way (Part 2) (July 22, 2014)- Algebraic thoughts on the threefold way and ten-fold way.
**Mike Shulman:**Pullbacks That Preserve Weak Equivalences (July 21, 2014)- On morphisms pullback along which preserves weak equivalences, and their name(s).
**Tom Leinster:**The Place of Diversity in Pure Mathematics (July 20, 2014)- Diversity is a canonical concept of pure mathematics.
**John Baez:**The Ten-Fold Way (Part 1) (July 19, 2014)- There are 10 associative real super-division algebras, 10 classical families of compact symmetric spaces, 10 ways that Hamiltonians can get along with time reversal and charge conjugation symmetry… and even 10 dimensions of spacetime in string theory! Are these related? Yes!
**Tom Leinster:**Math and Mass Surveillance: A Roundup (July 15, 2014)- List of all the posts so far on mathematics and mass surveillance.
**John Baez:**Describing PROPs Using Generators and Relations (July 10, 2014)- What’s a good reference for ‘presentations’ of symmetric monoidal category?
**Tom Leinster:**The Categorical Origins of Lebesgue Integration (July 8, 2014)- A simple universal property characterizing the space L^1 of Lebesgue-integrable functions.
**Mike Shulman:**The Linearity of Traces (July 1, 2014)- A success story for applied category theory: formal results about bicategorical trace imply the linearity of traces under absolute colimits.
**Emily Riehl:**Kan Extension Seminar Talks at CT2014 (June 28, 2014)- Kan extension seminar talks at CT2014.
**Emily Riehl:**Enriched Indexed Categories, Again (June 27, 2014)- Describes “enriched indexed categories” which generalize and unify enriched, internal, and indexed categories following a paper of Mike Shulman.
**Emily Riehl:**Categorical Homotopy Theory (June 7, 2014)- Describes the content of the new book “Categorical Homotopy Theory” just published by Cambridge University Press.
**Tom Leinster:**Should Mathematicians Cooperate with GCHQ? Part 3 (June 5, 2014)- Mathematicians: let’s have a properly evidence-based discussion of the ethics of working for the NSA, GCHQ and other intelligence agencies.
**Emily Riehl:**Codescent Objects and Coherence (June 2, 2014)- Summarizes the approach to coherence theorems expressed as a rectification for pseudoalgebras for a 2-monad using codescent objects, following Steve Lack.
**John Baez:**Spans and the Categorified Heisenberg Algebra (Part 2) (May 25, 2014)- Watch a video of a talk about Jeffrey Morton and Jamie Vicary’s work on the categorified Heisenberg algebra.
**Simon Willerton:**Enrichment and the Legendre-Fenchel Transform II (May 22, 2014)- Get the second installment of how Legendre-Fenchel duality is an example of the profunctor nucleus construction.
**Emily Riehl:**Classifying by Generalizing: The Theory of Accessibility Relative to a Limit Doctrine (May 20, 2014)- Summarizes “A Classification of Accessible Categories,” which characterizes those categories that are locally presentable or accessible relative to a sound limit doctrine.
**Tom Leinster:**Categories vs. Algebras (May 14, 2014)- Understanding the category algebra construction from representation theory.
**Tom Leinster:**Should Mathematicians Cooperate with GCHQ? Part 2 (April 30, 2014)- Reply to Richard Pinch’s article on GCHQ in the London Mathematical Society newsletter.
**David Corfield:**Homotopy Type Theory: Unified Foundations of Mathematics and Computation (April 29, 2014)- Funding for homotopy type theory
**Emily Riehl:**On Two-Dimensional Monad Theory (April 28, 2014)- Describes the approach to two-dimensional universal algebra taken in the paper of Blackwell, Kelly, and Power on two-dimensional monad theory.
**Tom Leinster:***New Scientist*Article (April 27, 2014)- Mathematicians should not cooperate with agencies of mass surveillance.
**John Baez:**Finite Products Theories (April 20, 2014)- I need a slight generalization of a classic result on algebraic theories. You may know who proved it!
**Emily Riehl:**Elementary Observations on 2-Categorical Limits (April 18, 2014)- Describes Kelly’s “Elementary observations on 2-categorical limits” and the general theory of weighted limits and colimits, which are described here in a special case.
**Simon Willerton:**Enrichment and the Legendre–Fenchel Transform I (April 16, 2014)- Remind yourself about basics of the Legendre–Fenchel transform.
**Simon Willerton:**universo.math (April 14, 2014)- Check out a new Spanish language maths magazine
**Simon Willerton:**The Modular Flow on the Space of Lattices (April 7, 2014)- See Bruce tell of Ghys’ modular knots.
**Emily Riehl:**On a Topological Topos (April 7, 2014)- A survey of Peter Johnstone’s “On a Topological Topos”.
**Tom Leinster:**Big Data Power (April 1, 2014)- Guest post by Nils Carqueville and Daniel Murfet: the big data generated by whole-population surveillance.
**John Baez:**Operads and Trees (March 31, 2014)- It would be nice if someone had already proved these conjectured results relating trees and operads.
**Tom Leinster:**Fourier Series and Flipped Classrooms (March 30, 2014)- Moving beyond traditional lecturing.
**Emily Riehl:**An Exegesis of Yoneda Structures (March 24, 2014)- Motivates the notion of Yoneda structure as an expression of basic notions of category theory in a natural 2-categorical language.
**John Baez:**Translating Grothendieck’s Biography into English (March 18, 2014)- Leila Schneps is raising funds for an English translation of volume 3 of a biography of Grothendieck.
**Simon Willerton:**Fuzzy Logic and Enriching Over the Category [0,1] (March 15, 2014)- Watch me try to understand fuzzy logic from an enriched category theory perspective
**Emily Riehl:**Review of the Elements of 2-Categories (March 9, 2014)- Summarizes the paper of Kelly and Street “Review of the elements of 2-categories” containing common background material for papers in the Sydney Category Seminar Lecture Notes 420
**Tom Leinster:**Operads of Finite Groups (March 5, 2014)- Guest post by Nick Gurski.
**Tom Leinster:**Should Mathematicians Cooperate with GCHQ? (March 2, 2014)- Article for the LMS Newsletter on an ethical decision facing mathematicians.
**Emily Riehl:**Metric Spaces, Generalized Logic, and Closed Categories (February 21, 2014)- Summarizes Lawvere’s “Metric Spaces, Generalized Logic, and Closed Categories” which explores applications of enriched category theory to metric topology.
**John Baez:**Relative Entropy (February 13, 2014)- John Baez and Tobias Fritz have a new category-theoretic characterization of relative entropy.
**David Corfield:**Call for Papers: Symposium on the Foundations of Mathematics (February 12, 2014)- conference call for papers
**Tom Leinster:**The Deteriorating Relationship Between Academics and the NSA (February 11, 2014)- NSA rejects academics, academics reject NSA.
**John Baez:**Network Theory Talks at Oxford (February 10, 2014)- There will be 4 talks on “Network Theory” and one on “Operads and the Tree of Life” at the University of Oxford.
**Simon Willerton:**Galois Correspondences and Enriched Adjunctions (February 5, 2014)- Translate from category theory to order theory
**Emily Riehl:**Categories of Continuous Functors (February 5, 2014)- Summarizes the orthogonal subcategory problem and its solution, presented in Freyd and Kelly’s “Categories of continuous functors I”
**Emily Riehl:**An Emerging Pattern in Algebra and Topology II (February 2, 2014)- An emerging pattern in algebra and topology leads to a new notion of finitely generated FI-modules, which capture the representation stable sequences that arise in practice.
**Emily Riehl:**An Emerging Pattern in Algebra and Topology I (February 1, 2014)- An emerging pattern in algebra and topology leads to a new notion of representation stability, inspired by homological stability.
**Tom Leinster:**Weightings for Compact Metric Spaces (January 30, 2014)- Guest post by Mark Meckes, relating new geometric invariants to old.
**John Baez:**Tensor Categories and TQFT in Erlangen (January 30, 2014)- There will be a meeting on tensor categories and topological field theory in Erlangen on March 4-6, 2014.
**Emily Riehl:**Formal Theory of Monads (Following Street) (January 27, 2014)- Summarizes and contextualizes Street’s 1972 paper “The formal theory of monads.”
**Tom Leinster:**The Magnitude of a Graph (January 22, 2014)- An analogue of cardinality for graphs, associating to each graph a rational function.
**Tom Leinster:**Hilarious Takedown of Bonkers Maths in Top Psychology Journal (January 21, 2014)- Make sure you have at least 2.9013 positive emotions for each negative one.
**John Baez:**Wormholes and Entanglement (January 19, 2014)- The formation of a wormhole and the creation of a particle-antiparticle pair are closely related processes in a 3d TQFT. This sheds some light on how wormholes and entanglement are similar.
**Tom Leinster:**The Electronic Frontier Foundation at the Joint Meetings (January 13, 2014)- An antidote to NSA recruitment at the Joint Math Meetings in Baltimore this week.
**Emily Riehl:**An Elementary Theory of the Category of Sets (January 12, 2014)- Summarizes and contextualizes Lawvere’s 1965 paper “An elementary theory of the category of sets.”
**Tom Leinster:**Academics Against Mass Surveillance (January 8, 2014)- Add your name to a public declaration against the systematic recording of our daily activities.
**Simon Willerton:**Ends (January 5, 2014)- End your ignorance of ends!
**Tom Leinster:**The Long Grind of Writing a Book (December 25, 2013)- Why it so painful?
**Tom Leinster:**Commuting Limits and Colimits over Groups (December 21, 2013)- An apparently new limit-colimit commutation theorem.
**Tom Leinster:**A Technical Innovation (December 10, 2013)- How to view the source TeX of mathematical expressions on this site.
**Emily Riehl:**Kan Extension Seminar applications (November 25, 2013)- A friendly reminder: applications for the Kan Extension Seminar are due at the end of the week. More information can be found in the initial announcement and on the seminar website. For those who don’t enroll, watch this space….
**Tom Leinster:**The 8th Scottish Category Theory Seminar (November 21, 2013)- Announcement of the 8th Scottish Category Theory Seminar
**David Corfield:**The Covariance of Coloured Balls (November 19, 2013)- covariance through coloured balls
**John Baez:**Categories for the Working Philosopher (November 18, 2013)- Elaine Landry is editing a book called
*Categories for the Working Philosopher*. **Tom Leinster:**Four New Talks (November 12, 2013)- The talks I gave in October.
**Tom Leinster:**Severing Ties with the NSA (November 10, 2013)- Should mathematical societies refuse to cooperate with the NSA and GCHQ?
**David Corfield:**The HoTT Approach to Physics (October 30, 2013)- Urs’s book published
**David Corfield:**Jet Categories at the nForum (October 21, 2013)- nLab activity
**Emily Riehl:**Announcing the Kan Extension Seminar (October 16, 2013)- Announcement for an online graduate reading course in category theory.
**Tom Leinster:**Unexpected Connections (October 6, 2013)- What are your favourite examples of unexpected connections in mathematics?
**David Corfield:**Witten Looking Anew at the Jones Polynomial (October 3, 2013)- Witten at the Clay Meeting
**Tom Leinster:**Who Ordered That? (October 2, 2013)- A very odd and elementary conjecture of Kontsevich has now been proved.
**John Baez:**Clay Mathematics Conference (September 24, 2013)- There is a Clay Research Conference from September 29 to October 4, 2013 at Oxford University.
**Tom Leinster:**The Definition of Graph is Classified (September 24, 2013)- Top Secret: a graph is a set of objects linked by edges.
**Simon Willerton:**New Teaching Method Improves Math Skills (September 19, 2013)- A novel teaching method involves figuring out what students do and don’t understand
**John Baez:**Integral Octonions (Part 6) (September 17, 2013)- Learn about the Cayley integers, a lattice in the octonions that’s closed under multiplication.
**Emily Riehl:**Mapping (Co)cylinder Factorizations via the Small Object Argument (September 16, 2013)- The enriched version of the algebraic small object argument produces the mapping (co)cylinder factorizations for chain complexes of modules over a commutative ring.
**Simon Willerton:**Classical Dualities and Formal Concept Analysis (September 12, 2013)- Find out what algebraic varieties, convex sets, linear subspaces, real numbers, logical theories and extension fields have in common with formal concepts.
**David Corfield:**Good Mathematics (September 12, 2013)- Conference announcement on values in mathematics
**Alexander Hoffnung:**CRM (Centre de Recherches Mathématiques, Montreal) Thematic Semester “New Directions in Lie Theory (September 10, 2013)- Much like folks around here, Alistair Savage has been thinking about categorification for some time now and has together with Erhard Neher announced an upcoming opportunity for graduate students and postdocs to take part in a winter school as…
**John Baez:**Integral Octonions (Part 5) (September 3, 2013)- Let’s count the number of vertices, edges, triangles, tetrahedra and so on in the root polytope of E
_{8}! **Simon Willerton:**Formal Concept Analysis (September 2, 2013)- Have a peek at the notion of formal concept analysis
**Tom Leinster:**Linear Operators Done Right (August 22, 2013)- On the very useful analogy between complex numbers and linear operators.
**Simon Willerton:**The Nucleus of a Profunctor: Some Categorified Linear Algebra (August 19, 2013)- Watch some linear algebra being categorified.
**Tom Leinster:**The Shannon Capacity of a Graph, 2 (August 18, 2013)- Connections between Shannon capacity, maximum diversity, and magnitude.
**Simon Willerton:**Oxford Workshop in Quantum Mathematics and Computation (August 15, 2013)- Read about the October workshop.
**Tom Leinster:**The Shannon Capacity of a Graph, 1 (August 10, 2013)- When you spell a word down the phone, some letters are more easily confused than others. What’s a good measure of your communication power that takes these confusions into account?
**Mike Shulman:**What is a Spectral Sequence? (August 4, 2013)- Spectral sequences are to long exact sequences as iterated extensions are to extensions.
**John Baez:**Integral Octonions (Part 4) (August 3, 2013)- More on semiregular polytopes and honeycombs related to the Cayley integer octonions.
**John Baez:**Spans and the Categorified Heisenberg Algebra (August 1, 2013)- Some talks on spans of groupoids, symmetric monoidal bicategories and how these structures shed new light on the combinatorics of quantum mechanics.
**Tom Leinster:**Keep Calm and Carry One (July 30, 2013)- How New Zealand schoolchildren aren’t learning to add.
**John Baez:**Integral Octonions (Part 3) (July 29, 2013)- Let’s take a look at the E
_{8}lattice and the Lie algebra by the same name. **David Corfield:**Minhyong Kim in The Reasoner (July 28, 2013)- Interview with Minhyong Kim
**Mike Shulman:**Cohomology Detects Failures of Classical Mathematics (July 28, 2013)- The axiom of choice (including the law of excluded middle) is equivalent to the statement that H^1 of all discrete sets with values in all groups vanishes.
**John Baez:**Integral Octonions (Part 2) (July 25, 2013)- Belinskii, Khalatnikov and Lifshitz described homogeneous cosmologies that become chaotic as you go back toward the Big Bang. What does this have to do with octonionic integers?
**Mike Shulman:**Higher Compactness (July 24, 2013)- Compactness and local compactness are the first rungs on infinite hierarchies of conditions that can be defined using higher toposes.
**John Baez:**Integral Octonions (Part 1) (July 23, 2013)- Learn how integral octonions are connected to the Weyl group of the hyperbolic Kac–Moody algebra E
_{10}. **Emily Riehl:**A Weighted Limits Proof of Monadicity (July 10, 2013)- The proof of Beck’s monadicity theorem is “all in the weights.”
**Mike Shulman:**Synthetic ∞-Groupoid Theory (July 10, 2013)- Homotopy type theory can be viewed as a synthetic theory of infinity-groupoids.
**Tom Leinster:**The First Commutative Diagram? (July 8, 2013)- A commutative square from 1919, in a book by Bertrand Russell.
**Tom Leinster:**Whitney Twists (July 7, 2013)- Q. Is the magnitude of a graph invariant under Whitney twists? A. Sometimes.
**Mike Shulman:**The HoTT Book (June 20, 2013)- The HoTT Book is out!
**John Baez:**Quasicrystals and the Riemann Hypothesis (June 14, 2013)- Freeman Dyson asserted a relation between quasicrystals and the Riemann Hypothesis. What does it amount to, actually?
**John Baez:**The Selected Papers Network (June 14, 2013)- Given that social networks already exist, all we need for truly open scientific communication is a convention on a consistent set of tags and IDs for discussing papers. Christopher Lee has developed software that makes this work.
**John Huerta:**G_{2}and the Rolling Ball (June 7, 2013)- I just returned from a month at Hong Kong University, visiting James Fullwood, an algebraic geometer who likes to think about the mathematics of string theory. There, I gave a colloquium on G2 and the rolling ball, a paper…
**Simon Willerton:**Torsors and Enriched Categories (June 3, 2013)- Read about a different take on torsors
**Tom Leinster:**Welcome, Emily (June 1, 2013)- Emily Riehl joins us as a host on this blog.
**Simon Willerton:**Philosophy Talks in Oxford (May 22, 2013)- Read about recent talks by Kremnitzer and Corfield
**David Corfield:**In the News (May 22, 2013)- article on category theory
**John Baez:**Semantics of Proofs in Paris (May 17, 2013)- There’s a program on “Semantics of Proofs and Certified Mathematics” in Paris from April 22nd to July 11tth, 2014.
**Mike Shulman:**The Propositional Fracture Theorem (May 16, 2013)- The gluing construction on topological spaces and topoi seems like a “propositional” version of the fracture theorems from algebraic topology.
**Tom Leinster:**Bounded Gaps Between Primes (May 14, 2013)- Report on recent work of Yi Tang Zhang, taking us a step closer to a proof of the Twin Primes Conjecture.
**John Baez:**How Does Applied Math Impact Foundations? (April 30, 2013)- This weekend I’m giving a talk on “The Foundations of Applied Mathematics”. It’s mostly about how the widespread use of diagrams in engineering, biology, and the like may force us to think about math in new ways, at least…
**Simon Willerton:**The Michael and Lily Atiyah Portrait Gallery (April 24, 2013)- View the gallery online.
**David Corfield:**Modal Types (April 23, 2013)- Applying modalities to types
**Simon Willerton:**Koudenburg on Algebraic Weighted Colimits (April 17, 2013)- Read about my student’s thesis.
**Mike Shulman:**Scones, Logical Relations, and Parametricity (April 17, 2013)- The category-theoretic scone or “gluing construction” packages the type-theorist’s method of “logical relations” to prove canonicity and parametricity properties of type theory.
**Tom Leinster:**Tutte Polynomials and Magnitude Functions (April 16, 2013)- The magnitude function is a mysterious invariant of graphs. It’s closely related to invariants that have proven meaningful and useful in other branches of mathematics. But uncovering its meaning for graphs poses a challenge.
**Tom Leinster:**Colouring a Graph (April 12, 2013)- A categorical perspective on graph colourings, and a fundamental gap in our knowledge.
**John Baez:**Category-Theoretic Foundations in Irvine (April 5, 2013)- You can now see the list of talks at Category-Theoretic Foundations of Mathematics Workshop at the Department of Logic and Philosophy of Science of U.C. Irvine, May 4-5, 2013.
**John Baez:**Iterating the Free Monoid Construction (April 3, 2013)- If you cube the free monoid monad, do you get back where you started… in some weak sense?
**John Baez:**The Hipparchus Operad (April 1, 2013)- It seems the Schröder–Hipparchus numbers are the numbers of operations in an operad that was first discovered around 100 BC.
**Tom Leinster:**The Convex Magnitude Conjecture (March 30, 2013)- An unsolved problem about measure of convex sets, arising from category theory.
**David Corfield:**Categories for the Working Metaphysician (March 27, 2013)- conference on category theory and philosophy
**Simon Willerton:**Project Scheduling and Copresheaves (March 24, 2013)- Find out what PERT graphs have to do with enriched categories
**Tom Leinster:**Five Positions at Edinburgh (March 22, 2013)- Jobs at Edinburgh.
**Tom Leinster:**Mark on Magnitude (March 21, 2013)- Mark Meckes mouths magnificently on metric magnitude.
**Simon Willerton:**Lectureship in Sheffield (March 6, 2013)- Apply for a lectureship in Sheffield.
**Mike Shulman:**Category Theory in Homotopy Type Theory (March 5, 2013)- Doing category theory in homotopy type theory allows us to satisfy the principle of equivalence automatically.
**Simon Willerton:**Spivak on Category Theory (March 3, 2013)- See a new book for scientists about category theory that’s been put on the arXiv.
**David Corfield:**Have You Left Off Beating Your Wife? (February 20, 2013)- Continuing the series of what type theory can do for philosophy, let us take a look at the infamous question of the title. To forestall criticism, let me say straight away that I’m not proposing in these posts that…
**John Baez:***M*_{13}(February 10, 2013)- You can use groupoids to give a nice description of the Mathieu group $M_{12}$!
**David Corfield:**The Title of This Post is False (February 6, 2013)- type theory and the liar paradox
**Mike Shulman:**Presentations and Representations in Foundations (February 4, 2013)- With a type-theoretic foundational system, we can describe constructions on objects with internalized universal properties, including the hom-functors of higher groupoids.
**David Corfield:**The King of France (January 30, 2013)- Bringing dependent type theory to philosophy
**Tom Leinster:**7th Scottish Category Theory Seminar (January 29, 2013)- Conference announcement.
**Tom Leinster:**This Week’s Finds at 20 (January 26, 2013)- The 20th birthday of John’s column.
**Simon Willerton:**Tight spans, Isbell completions and semi-tropical modules (January 20, 2013)- See how these three things are related.
**Tom Leinster:**Carleson’s Theorem (January 17, 2013)- The twisty turny history of beliefs about pointwise convergence of Fourier series.
**Mike Shulman:**The Universal Property of Categories (January 16, 2013)- Using the theory of enriched bicategories, we can exhibit the construction of enriched categories as a free cocompletion in a world of equipments.
**Mike Shulman:**Two Dimensional Monadicity (January 11, 2013)- An extension of the monadicity theorem using F-categories to detect lax, pseudo, and colax morphisms provides new insight into the fundamental nature of such weak morphisms.
**Mike Shulman:**From Set Theory to Type Theory (January 7, 2013)- Starting from set theories like ZFC and ETCS, trying to take the best of both worlds leads us naturally to univalent type theory.
**Urs Schreiber:***A Higher Stacky Perspective On Chern-Simons Theory*(December 28, 2012)- A contribution for a book on mathematical aspects of QFT, on extended (“multi-tiered”) prequantum Chern-Simons theory formulated in terms of higher geometry.
**Tom Leinster:**Category Theory 2013 (December 24, 2012)- Announcement of Category Theory 2013, Sydney.
**Mike Shulman:**Names For Equivalences (December 20, 2012)- What names would you give to all these equivalent notions of “equivalence”?
**John Baez:**Category-Theoretic Foundations in Irvine (December 18, 2012)- There will be a conference on Category-Theoretic Foundations of Mathematics at the Department of Logic and Philosophy of Science, U.C. Irvine on May 4-5, 2013.
**Tom Leinster:**Rethinking Set Theory (December 18, 2012)- Expository paper on set theory done categorically.
**Mike Shulman:**Enriched Indexed Categories (December 18, 2012)- Enriched indexed categories, indexed over a base category and enriched over an indexed monoidal category, simultaneously generalize indexed categories, internal categories, and enriched categories.
**Mike Shulman:**The Additivity of Traces in Stable Monoidal Derivators (December 14, 2012)- Stable monoidal derivators give a new, easier, and more flexible context for proving the additivity of traces and generalizing it.
**Mike Shulman:**Universe Polymorphism and Typical Ambiguity (December 9, 2012)- “Universe polymorphism” and “typical ambiguity” are fancy words for simple language.
**Tom Leinster:**Bob on TAC (December 8, 2012)- Bob Rosebrugh’s article about Theory and Applications of Categories in the Notices of the AMS.
**Tom Leinster:**Research Fellowships in Edinburgh (December 7, 2012)- Five postdoc positions available.
**David Corfield:**What Can Category Theory Do For Philosophy? (December 6, 2012)- A possible philosophical meeting on category theory
**Tom Leinster:**Almost All of the First 50 Billion Groups Have Order 1024 (November 28, 2012)- More than 99 per cent of the groups of order at most 2000 have order 1024.
**Urs Schreiber:***TwoVect*(November 19, 2012)- Jamie Vicary on a software for handling 2-categorical linear algebra.
**Mike Shulman:**Freedom From Logic (November 9, 2012)- Towards a way of doing homotopy type theory informally, and a new opinion regarding propositions as types.
**David Corfield:**Back in Business (November 9, 2012)- Sorry about that interruption in service. We seem to be back in business, but I fear we may have lost some of the more recent comments….
**Tom Leinster:**Parametrized Mates and Multivariable Adjunctions (November 1, 2012)- Guest post by Emily Riehl.
**Tom Leinster:**The Zorn Identity (October 26, 2012)- Zorn’s lemma is trivially equivalent to a theorem provable without the axiom of choice.
**Mike Shulman:**Object Classifiers and (∞,1)-Quasitoposes (October 18, 2012)- A preprint by Gepner and Kock about object classifiers in (infinity,1)-toposes and (infinity,1)-quasitoposes.
**Tom Leinster:**The Curious Dependence of Set Theory on Order Theory (October 13, 2012)- Is set theory as dependent on order theory as it seems to be, and if so, why?
**Tom Leinster:**Analysis Fellowship in Edinburgh (October 11, 2012)- Job ad.
**Urs Schreiber:**Tangency (October 9, 2012)- Thoughts by Andrew Stacey on the notion of tangency.
**Tom Leinster:**Symmetric Monoidal Bicategories (October 4, 2012)- Guest post by Nick Gurski.
**Tom Leinster:**Where Do Ultraproducts Come From? (September 30, 2012)- Are ultraproducts categorically inevitable?
**Tom Leinster:**Where Do Linearly Compact Vector Spaces Come From? (September 25, 2012)- What should it mean for a topological vector space to be compact?
**Tom Leinster:**Where Do Ultrafilters Come From? (September 23, 2012)- The notion of ultrafilter is inherent in the notion of finiteness of a set.
**Tom Leinster:**Where Do Monads Come From? (September 20, 2012)- Monads don’t just come from adjunctions.
**Mike Shulman:**Gauge Spaces and the Stone-Cech Compactification (September 14, 2012)- Some expository notes developing the Stone-Cech compactification using gauge spaces instead of topological spaces.
**John Huerta:**Introducing the GR & QFT seminar (September 13, 2012)- Here at the Australian National University, I’ve started running a seminar with Mathew Langford. Mat is a PhD student working on geometric analysis, specifically something called extrinsic curvature flow, but we both love mathematical physics, so we decided to…
**David Corfield:**The Ax-Grothendieck Theorem According to Category Theory (September 10, 2012)- Can category theory account for Ax-Grothendieck
**Mike Shulman:**Every Functor is a (Co)Reflection (September 10, 2012)- Every functor can be made into a reflection or coreflection by changing its domain. There is a formal and unenlightening proof, but for some examples we can find “natural” proofs which are actually informative.
**Urs Schreiber:**General Covariance in Homotopy Type Theory (September 8, 2012)- A natural formalization of generally covariant field theory in the formal language of homotopy type theory.
**Simon Willerton:**The Spread of a Metric Space (September 5, 2012)- Read how this notion of size for metric spaces has some interesting properties.
**Urs Schreiber:***From Poisson To String Geometry*(September 4, 2012)- A conference in Erlangen on Poisson geometry, String geometry, and some relations.
**David Corfield:**Urs Schreiber in The Reasoner (August 30, 2012)- Discussing Urs Schreiber’s interview in The Reasoner.
**David Corfield:**Directed Homotopy at Liverpool (August 30, 2012)- PhD position at Liverpool in higher category theory
**Tom Leinster:**Jech (1973), page 118 (August 29, 2012)- An amusing page of Thomas Jech’s book “The Axiom of Choice”.
**Mike Shulman:**Brown representability (August 24, 2012)- Surprisingly enough, Brown’s representability theorem is false for nonconnected or unpointed spaces.
**Urs Schreiber:***Quantum Gauge Field Theory in Cohesive Homotopy Type Theory*(August 20, 2012)- A survey article on the formalization of aspects of quantum gauge field theory in the formal language of cohesive homotopy type theory.
**Urs Schreiber:***Strings and Automorphic Forms in Topology*(August 13, 2012)- A conference on mathematical issues in higher string geometry.
**David Corfield:**Categorical Unification (August 8, 2012)- Postdocs in Categorical Unification
**Urs Schreiber:***Extended higher Chern-Simons theories*(July 20, 2012)- A discussion of “extended” or “multi-tiered” prequantum higher-dimensional higher gauge theories of Chern-Simons type.
**Mike Shulman:**Zero-Determinant Strategies in the Iterated Prisoner’s Dilemma (July 19, 2012)- A revolutionary paper by Press and Dyson describes surprising and exciting new strategies for the Iterated Prisoner’s Dilemma – but they don’t “beat” TIT FOR TAT unless we change the way we measure victory.
**Urs Schreiber:**Notes from*String-Math 2012*(July 16, 2012)- Notes from String-Math 2012, a conference on mathematical aspects of string theory.
**John Baez:**Morton and Vicary on the Categorified Heisenberg Algebra (July 10, 2012)- Jeffrey Morton and Jamie Vicary have a new paper on the categorification of quantum mechanics using spans of groupoids.
**Urs Schreiber:**Geometric Quantization on Moduli ∞-Stacks (July 9, 2012)- A talk on higher geometric quantization on higher moduli stacks.
**Urs Schreiber:**Flat Ehresmann connections in Cohesive HoTT (June 28, 2012)- The formalization of flat Ehresmann connections in higher geometry.
**Urs Schreiber:**Principal ∞-Bundles – general theory and presentations (June 25, 2012)- Notes on the theory of principal and twisted bundles in higher geometry.
**Mike Shulman:**The Gamification of Higher Category Theory (June 18, 2012)- Coq, meet DragonBox.
**David Corfield:**Cohomology in Everyday Life (June 14, 2012)- intuitive examples of cohomology
**Mike Shulman:**Directed Homotopy Type Theory (June 7, 2012)- The internal homotopy type theory of the (infinity,1)-topos of simplicial objects gives us a “directed homotopy type theory” to talk about (infinity,1)-categories.
**John Baez:**Compact Closed Bicategories (June 6, 2012)- Mike Stay has a new paper on compact closed bicategories and their many applications.
**Mike Shulman:**Superextensive Sites (May 21, 2012)- “Superextensive sites” allow us to replace covering families by singleton covers… sometimes.
**Urs Schreiber:**Segal-Completeness and Univalence (May 18, 2012)- An observation about the relation between Segal-completeness in internal infinity-category theory and univalence in homotopy type theory.
**Tom Leinster:**Integrating Against the Euler Characteristic (May 17, 2012)- Euler characteristic is something like a measure. So what does integrating against it do?
**Mike Shulman:**Postulated Colimits and Absolute Colimits (May 14, 2012)- Absolute colimits, those which are preserved by any functor, are a special case of Anders Kock’s notion of “postulated colimit.”
**Urs Schreiber:**What is Homotopy Type Theory Good For? (May 10, 2012)- An example in mathematical string theory where appeal to homotopy type theory helps solve a subtle problem.
**Mike Shulman:**The Mysterious Nature of Right Properness (May 7, 2012)- What does right properness of a model category mean?
**Urs Schreiber:**Twisted Higher Bundles in Münster (May 5, 2012)- A workshop that happens to have some talks on twisted higher bundles.
**John Baez:**Quivering with Excitement (May 4, 2012)- Any projective variety is some Grassmannian of some quiver representation.
**Tom Leinster:**The Mathematics of Biodiversity (April 26, 2012)- Conference in Barcelona.
**Mike Shulman:**PSSL 93 trip report (April 25, 2012)- A report on several talks from the PSSL 93 in Cambridge.
**Urs Schreiber:**Principal ∞-Bundles (April 23, 2012)- A preview of an article on principal bundles in higher geometry.
**Tom Leinster:**10,000 (April 23, 2012)- News on Elsevier boycott.
**Urs Schreiber:***Higher Structures in Topology and Geometry VI*(April 20, 2012)- Part VI of the series of workshops on Higher Structures in Göttingen.
**John Baez:**Teleparallel Gravity as a Higher Gauge Theory (April 20, 2012)- How to see Einstein’s old theory of ‘teleparallel gravity’ as a theory involving a 2-connection.
**David Corfield:**Mathematical Cultures (April 19, 2012)- Conference on mathematical cultures
**Urs Schreiber:**Research Network*String Geometry*(April 16, 2012)- We won a reaserch network grant on the topic of String Geometry, the higher differential geometry of differential string structures.
**Urs Schreiber:***CATS 4*(April 16, 2012)- Meeting in Luminy about Higher Categorical Structures and their Interactions with Algebraic Geometry, Algebraic Topology and Algebra
**Tom Leinster:**Wellcome Joins In (April 9, 2012)- I just looked at the front page of The Guardian, a quality British newspaper that’s especially widely read online… and I was amazed to see that their second-leading story was on The Cost of Knowledge: The news is that…
**Mike Shulman:**Exact completions and small sheaves (March 21, 2012)- Most kinds of exact completion, including categories of sheaves, are a special case of a reflection from certain sites into higher-ary exact categories.
**David Corfield:**Circles Disturbed (March 19, 2012)- Edited book on math and narrative appears
**Tom Leinster:**Reader Survey: log|*x*| +*C*(March 19, 2012)- On a worrying tradition in the teaching of calculus.
**Tom Leinster:**6th Scottish Category Theory Seminar (March 19, 2012)- Conference announcement.
**Urs Schreiber:***New Perspectives in Topological Field Theories*(March 16, 2012)- A conference in Hamburg on mathematical physics, with an emphasis on TQFT, representation theory and categorification.
**Mike Shulman:**The Multiplicativity of Fixed-Point Invariants (March 6, 2012)- A bicategorical point of view on fixed-point invariants implies a general multiplicativity theorem.
**Urs Schreiber:**Prequantization in Cohesive Homotopy Type Theory (February 27, 2012)- A proposal for axiomatics of geometric prequantization formulated in cohesive homotopy type theory.
**Tom Leinster:**Types, Homotopy and Univalent Foundations: Special Issue (February 26, 2012)- Announcement of a special journal issue on homotopy type theory
**Urs Schreiber:**ESI Program*K-Theory and Quantum Fields*(February 22, 2012)- A program at the ESI institute in Vienna including two weeks of lectures on cohomological structures in quantum field theory and string theory.
**Urs Schreiber:**In Göttingen (February 21, 2012)- Am in Göttingen, talking about higher differential geometry and higher gauge theory.
**John Baez:**Math 2.0 (February 16, 2012)- Welcome to Math 2.0: a forum for discussing an upgrade of the mathematical environment.
**Urs Schreiber:**Workshop on Formal Topology, Higher Dimensional Algebra, Categories and Types (February 15, 2012)- A workshop or two on topics in topology, type theory and homotopy theory.
**Urs Schreiber:**Modern Perspectives in Homotopy Theory (February 14, 2012)- A school on modern developments in homotopy theory.
**Urs Schreiber:**The Moduli 3-Stack of the C-Field (February 9, 2012)- An article discussing the moduli 3-stack of the C-field in 11-dimensional supergravity.
**John Baez:**The Cost of Knowledge (February 9, 2012)- An explanation of the Elsevier boycott, from the mathematician’s point of view.
**Alexander Hoffnung:**Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory (February 8, 2012)- Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory at the Fields Institute, May 7-9, 2012
**John Baez:**Good News (February 7, 2012)- Good news about category theory in Macquarie, Jeffrey Morton’s new position, and his work with Jamie Vicary on categorified quantum mechanics.
**John Baez:**Banning Elsevier (January 26, 2012)- Join the ban of Elsevier.
**Urs Schreiber:**Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory (January 21, 2012)- An article on 7-dimensional Chern-Simons theory of nonabelian String 2-form fields.
**Tom Leinster:**Vorsicht! Funktor! (January 15, 2012)- Sign in Germany.
**David Corfield:**Logic as Invariant-Theory (January 13, 2012)- Erlanger program for logic
**Mike Shulman:**Propositions as Some Types and Algebraic Nonalgebraicity (January 12, 2012)- Different ways to relate proofs with constructions in foundational theories, and why homotopy type theory makes the choice it does.
**Urs Schreiber:**Mathematical Aspects of String and M-Theory in Cambridge (January 11, 2012)- A workshop in Cambridge on mathematical aspects of string theory.
**Tom Leinster:**A Semigroup Approach to Finite Markov Chains (January 3, 2012)- Guest post by Benjamin Steinberg, proving the basic results on finite Markov chains using semigroup theory.
**John Baez:**The Café Will Be Closed 2 Days in January (December 30, 2011)- The Café will be closed for 48 hours starting at 6 pm CST on January 5th, 2012.
**Tom Leinster:**The Eventual Image, Part 2 (December 25, 2011)- In which categories do endomorphisms have eventual images? A solution to the problem posed in Part 1.
**Tom Leinster:**On the Law of Large Numbers (Such As 60) (December 20, 2011)- Alex Simpson’s work on randomness and locales.
**Tom Leinster:**What Do You Think of EPSRC Policy? (December 19, 2011)- Problems with EPSRC funding policy, and a call for comments by Burt Totaro and Michael Singer.
**David Corfield:**Back from NIPS 2011 (December 19, 2011)- Report on maching learning workshops
**Urs Schreiber:***4th Odense Winter School on Geometry and Mathematical Physics*(December 18, 2011)- A winter school on gauge theoretic aspects of theoretical physics.
**John Baez:**Categorifying Fractional Euler Characteristics (December 16, 2011)- Frenkel, Stroppel and Sussan have a way to define things with fractional Euler characteristics.
**John Baez:**What Might Be Done About High Prices of Journals? (December 12, 2011)- Do you have suggestions for how to lower the prices of mathematical journals?
**Mike Shulman:**Cellularity in Algebraic Model Structures (December 9, 2011)- Constructing algebraic versions of model structures, Quillen adjunctions, and monoidal model structures using cellularity of the generators.
**Tom Leinster:**The Eventual Image (December 8, 2011)- Three different categories exhibit very similar dynamical behaviour. Why?
**Alexander Hoffnung:**Spans in 2-Categories: A Monoidal Tricategory (December 7, 2011)- A paper constructing a monoidal tricategory of spans.
**Urs Schreiber:***Basic Ideas of Homotopy Axiomatic Cohesion*(December 6, 2011)- Slides for a course on basic ideas of axiomatic cohesion in homotopy type theory.
**Mike Shulman:**Reflective Subfibrations, Factorization Systems, and Stable Units (December 6, 2011)- When can a reflective subcategory be extended to a reflective subfibration or a stable factorization system?
**John Baez:**TeX in Gmail (December 5, 2011)- There’s a plugin for TeX in Gmail.
**Urs Schreiber:**Some Job Offers (December 5, 2011)- Below are various recent job offers in maths that colleagues are asking me to circulate. One in Liverpool on motives. One in Prague, on geometry and algebra. One in Erlangen, on higher categories and TQFT. One in Adelaide, on…
**John Baez:**The (∞,1)-Category of (∞,n)-Categories (November 30, 2011)- There’s a new paper by Barwick and Schommer-Pries on the (∞,1)-category of (∞n)-categories.
**Mike Shulman:**Internalizing the External, or The Joys of Codiscreteness (November 29, 2011)- An internal axiomatization of factorization systems, subtoposes, local toposes, and cohesive toposes in homotopy type theory, using “higher modality” and codiscreteness.
**Urs Schreiber:***Higher Structures Along The Lower Rhine*(November 29, 2011)- A workshop in Bonn on higher structures in geometry.
**Urs Schreiber:***Higher Structures 2011*(November 28, 2011)- The 2011 meeting of the annual workshop on
*Higher structures in mathematics and physics*. **Urs Schreiber:***Publications of the nLab*(November 23, 2011)- A web-based journal for publications in mathematics and physics for topics that are usefully viewed from the point of view of higher category theory and homotopy theory.
**Mike Shulman:**Discreteness, Concreteness, Fibrations, and Scones (November 22, 2011)- Categories of spaces with discrete and codiscrete objects are closely related to fibrations and opfibrations; the “scone” freely adds both codiscrete objects and cartesian arrows.
**Urs Schreiber:***Chern-Simons Terms on Higher Moduli Stacks*(November 15, 2011)- A talk on Chern-Simons functionals defined on higher moduli stacks of higher connections.
**Urs Schreiber:**HoTT Cohesion – Exercise II (November 9, 2011)- How to code in homotopy type theory the intrinsic definition of ordinary differential cohomology and the proof of its basic properties?
**Mike Shulman:**Productive Homotopy Pullbacks (November 7, 2011)- Certain squares built out of cartesian products are always homotopy pullbacks, in any derivator.
**Mike Shulman:**Traces in Indexed Monoidal Categories (November 4, 2011)- Indexed monoidal categories and their string diagrams provide a nice framework for generalized fixed point theory.
**Urs Schreiber:**HoTT Cohesion – Exercise I (November 2, 2011)- de Rham coefficient objects and Maurer-Cartan forms in cohesive homotopy type theory.
**Urs Schreiber:**WZW Models in a Cohesive ∞-Topos (October 31, 2011)- A talk on (higher dimensional) Wess-Zumino-Witten models internal to a cohesive infinity-topos.
**David Corfield:**Coalgebra Paper (October 31, 2011)- the appearance of my paper on coalgebra
**Urs Schreiber:***Prequantum Physics in a Cohesive ∞-Topos*(October 26, 2011)- Slides for a talk at “Quantum Physics and Logic 2011”.
**Tom Leinster:**Measuring Diversity (October 23, 2011)- A way of quantifying diversity, especially biological diversity, that takes similarity between species into account.
**John Baez:**A Math Puzzle Coming From Chemistry (October 23, 2011)- Try this graph theory puzzle coming from chemistry.
**Tom Leinster:**Do You Know This Idempotent? (October 19, 2011)- Every endomorphism of a finite set has a unique idempotent power. Can you describe it?
**Tom Leinster:**Spectra of Operators and Rings (October 16, 2011)- Connecting notions of spectra, and a question on characteristic polynomials.
**John Baez:**Weak Systems of Arithmetic (October 11, 2011)- There are lots of interesting axiom systems for arithmetic weaker than Peano arithmetic.
**Tom Leinster:**The 5th Scottish Category Theory Seminar (October 10, 2011)- Announcement of Scotcats 5.
**Urs Schreiber:**Theoretical Physics – Stack Exchange (October 5, 2011)- A new question-and-anwers site for research-level theoretical physics.
**David Corfield:**Zalamea on Sheaf Logic (October 3, 2011)- Zalamea seminar announcement
**John Baez:**The Inconsistency of Arithmetic (September 27, 2011)- Edward Nelson has claimed to prove the inconsistency of Peano Arithmetic.
**John Baez:**Division Algebras and Supersymmetry III (September 22, 2011)- John Huerta explains his new paper, which constructs some Lie 2-supergroups that are important in superstring theory.
**Urs Schreiber:**QVEST, Autumn 2011 (September 17, 2011)- The next meeting of our Quarterly Seminar on Topology and Geometry
**Tom Leinster:**Universal Measures (September 14, 2011)- Groemer’s integral theorem, recast as a universal property.
**David Corfield:**Who’s Your Friend? (September 12, 2011)- derive category up to equivalence or isomorphism?
**David Corfield:**Voevodsky on FOM (September 12, 2011)- Does Voevodsky understand Godel’s results?
**John Baez:**Mathematics of Planet Earth at Banff (September 8, 2011)- Let’s have a math conference in 2013 at Banff on the theme
*The Mathematics of Planet Earth*. **John Baez:**Math and Physics on Google+ (September 8, 2011)- Fun math and physics items at Google+.
**John Baez:**The Genius In My Basement (August 30, 2011)- There’s a new book about the mathematician Simon Norton.
**Tom Leinster:**Hadwiger’s Theorem, Part 2 (August 29, 2011)- Generalizing Hadwiger’s theorem to arbitrary metric spaces, especially R^n with the 1-norm.
**Tom Leinster:**Mixed Volume (August 26, 2011)- Several explanations of the mixed volume of convex bodies.
**David Corfield:**The Set-Theoretic Multiverse (August 23, 2011)- Why there are many universes of sets.
**Tom Leinster:**All Job Ads Should Be Like This (August 21, 2011)- Poetic ad for a job at the University of Strathclyde.
**Mike Shulman:**Fixed Point Indices for Groupoids (August 20, 2011)- Can you define the fixed-point index for finitely generated free groupoids in a purely categorical way?
**Alexander Hoffnung:**Fields Institute Workshop on Category Theoretic Methods in Representation Theory (August 18, 2011)- Workshop on category theoretic methods in representation theory.
**John Baez:**Klein 2-Geometry XII (August 17, 2011)- An attempt to see teleparallel gravity as a higher gauge theory.
**John Baez:**The Strangest Numbers in String Theory (August 15, 2011)- A
*Scientific American*article on normed division algebras, especially the octonions, and their role in string theory. **John Baez:**Geometries: Diffeomorphism Classes vs Quilts (August 13, 2011)- What follows is a guest post by Greg Weeks. If your memory extends back before the formation of this blog to the glory days of sci.physics.research, you should remember Greg….
**Urs Schreiber:**AKSZ-Models in Higher Chern-Weil Theory (August 6, 2011)- The class of AKSZ-sigma models has a natural origin in abstract higher Chern-Weil theory.
**Urs Schreiber:**AKSZ Sigma-Models (August 5, 2011)- The original definition of AKSZ sigma-model quantum field theories.
**Mike Shulman:**Coinductive Definitions (July 30, 2011)- Coinductive definitions are just as basic as inductive ones, and are an important tool for working with ω-categories.
**Urs Schreiber:**Local and Global Supersymmetry (July 27, 2011)- On the concept of supersymmetry and its experimental verification.
**Urs Schreiber:**Bohr Toposes (July 25, 2011)- The quantum phase space as a ringed topos.
**David Corfield:**Doctrinal and Tannakian Reconstruction (July 11, 2011)- Seeking what is common between Gabriel-Ulmer style dualities and Tannaka duality
**John Baez:**Operads and the Tree of Life (July 7, 2011)- Tomorrow I’m giving a talk about an operad that shows up in biology. I wrote my lecture notes in the form of a blog entry: John Baez, Operads and the tree of life, Combinatorics Seminar, Université du Québec à…
**Tom Leinster:**Definitions of Ultrafilter (July 2, 2011)- Several equivalent definitions of ultrafilter, including a particularly simple one that I’d like to find a reference for.
**Urs Schreiber:**Nikolaus on*Higher Categorical Structures in Geometry*(July 1, 2011)- Thomas Nikoluas’s PhD thesis on topics in higher geometry and gauge theory.
**Tom Leinster:**Hadwiger’s Theorem, Part 1 (June 29, 2011)- Expository account of Hadwiger’s theorem characterizing invariant valuations on convex sets.
**Urs Schreiber:***Towards the Mathematics of Quantum Field Theory*(June 21, 2011)- A book on modern attempts at formalizing quantization and quantum field theory.
**Mike Shulman:**Flat Functors and Morphisms of Sites (June 19, 2011)- Two classical notions of flat functor have a common generalization, leading to a better category of sites.
**David Corfield:**A Bit More About Entropy (June 17, 2011)- Entropy and geometry
**Urs Schreiber:**Differential Characteristic Cocycles (June 16, 2011)- Notes on refined differential characteristic maps.
**Tom Leinster:**The Magnitude of an Enriched Category (June 13, 2011)- An analogue of cardinality for enriched categories.
**John Baez:**Christopher Walker on Hall Algebras (June 11, 2011)- Christopher Walker has successfully defended his thesis,
*A Categorification of Hall Algebras*. **Urs Schreiber:**String Topology Operations as a Sigma-Model (June 6, 2011)- The origin of string topology operation in extended QFT and the derivation of the spaces of states from higher line bundles.
**John Baez:**Category-Theoretic Characterizations of Entropy III (June 5, 2011)- Now we’re trying to generalize our characterization of entropy from the classical to the quantum case.
**Tom Leinster:**Möbius Inversion for Categories (May 31, 2011)- The relationships between Euler characteristic and different kinds of Möbius inversion.
**Tom Leinster:**An Operadic Introduction to Entropy (May 18, 2011)- Summary of recent results on entropy, and introduction to entropy for the categorically minded.
**Urs Schreiber:**∞-Dijkgraaf-Witten Theory (May 12, 2011)- In another thread Tom Leinster would like to learn what a sigma-model in quantum field theory is. Here I want to explain this in a way that will make perfect sense to Tom, and hopefully even intrigue him. To…
**David Corfield:**Making Things Simpler by Duality (May 12, 2011)- How should we understand the simplification introduced by duality?
**Urs Schreiber:**QVEST, Summer 2011 (May 11, 2011)- The fourth QVEST meeting.
**Tom Leinster:**Entropies*vs.*Means (May 10, 2011)- Connections between the two concepts.
**John Baez:**Category-Theoretic Characterizations of Entropy II (May 8, 2011)- More work on trying to understand entropy in category-theoretic terms.
**John Baez:**Category-Theoretic Characterizations of Entropy (May 1, 2011)- Tobias Fritz, Tom Leinster and John Baez are pondering category-theoretic characterizations of entropy.
**John Baez:**Higher Categories for Concurrency (April 29, 2011)- Mike Stay on a bicategorical model of Boudol’s ‘blue calculus’.
**Tom Leinster:***The Colours of Infinity*— A Review (April 26, 2011)- Review of a popular mathematics book.
**Mike Shulman:**Homotopy Type Theory, VI (April 24, 2011)- A homotopical extension of the notion of “inductive definition” allows us to construct CW complexes and mimic other constructions from homotopy theory in type theory.
**Urs Schreiber:**Workshop on Higher Gauge Theory, TQFT and Categorification in Cardiff (April 21, 2011)- A workshop in Cardiff on higher gauge theory and quantum field theory.
**John Baez:**Higher Structures in China II (April 21, 2011)- There is a conference Higher structures in China II taking place at Jilin University on August 8th to 10th, 2011.
**John Baez:**Quantum Theory and Gravitation in Zurich (April 17, 2011)- There’s a conference on Quantum Theory and Gravitation at the ETH in Zurich, June 14–24, 2011.
**David Corfield:**Report on Peter Freyd’s Lecture (April 16, 2011)- Freyd on philosophy of mathematics.
**Urs Schreiber:**Segal on*Three roles of quantum field theory*(April 14, 2011)- Graeme Segal lectures on quantum field theory at the HIM institute in Bonn.
**Mike Shulman:**Enhanced 2-categories and Limits for Lax Morphisms (April 13, 2011)- What 2-categorical limits are there in the 2-category of algebras and lax morphisms for a 2-monad?
**John Baez:**The Three-Fold Way (Part 6) (April 13, 2011)- See how the Threefold Way solves two ‘problems’ with real and quaternionic quantum theory.
**Urs Schreiber:**Axioms for Infinitesimal Cohesion (April 12, 2011)- Simple homotopy theoretic axioms for differential geometry/cohomology with explicit (synthetic) infinitesimals.
**Mike Shulman:**Homotopy Type Theory, V (April 12, 2011)- What’s still missing to make homotopy type theory into a complete internal language for (∞,1)-topoi?
**Tom Leinster:**The 4th Scottish Category Theory Seminar (April 7, 2011)- Seminar announcement.
**Mike Shulman:**Homotopy Type Theory, IV (April 5, 2011)- Voevodsky’s “univalence axiom” for universes in homotopy type theory, and the equivalent principle of “equivalence induction.”
**David Corfield:**Category Theoretic Modal Logic (April 4, 2011)- Category theoretic ideas on modal logic
**Alexander Hoffnung:**A Tetracategory of Spans (or, What Is a Monoidal Tricategory?) (March 30, 2011)- A paper on monoidal tricategories of spans.
**Tom Leinster:**Which Graphs Can be Given a Category Structure? (March 27, 2011)- A report on Samer Allouch’s thesis.
**David Corfield:**An Anti-Philosophy of Mathematics (March 25, 2011)- Announcing a Peter Freyd lecture
**Mike Shulman:**Homotopy Type Theory, III (March 24, 2011)- Some more discussion of the notions of “equivalence” in homotopy type theory.
**John Baez:**Higher Gauge Theory, Division Algebras and Superstrings (March 22, 2011)- See the slides for a talk explaining John Huerta’s work on higher gauge theory, division algebras and superstrings.
**Mike Shulman:**Homotopy Type Theory, II (March 18, 2011)- Continuing from last time, we define sets and logic in terms of homotopy type theory.
**Mike Shulman:**Homotopy Type Theory, I (March 11, 2011)- An overview of the correspondence between homotopical structures and intensional type theory.
**John Baez:**A Categorified Supergroup for String Theory (March 11, 2011)- John Huerta has used the octonions to construct a Lie 3-supergroup that describes the parallel transport of 2-branes in 11 dimensions.
**David Corfield:**Category Theory and Metaphysics (March 9, 2011)- How does category theory help metaphysics?
**David Corfield:**Symposium: Sets Within Geometry (March 7, 2011)- There is to be a Symposium – Sets Within Geometry – held in Nancy, France on 26-29 July, 2011. Confirmed speakers are: FW Lawvere (Buffalo), Yuri I. Manin (Bonn and IHES), Anders Kock (Aarhus), Christian Houzel (Paris), Colin McLarty…
**John Baez:**Liang Kong on Levin-Wen Models (March 7, 2011)- Liang Kong is using tricategories in condensed matter physics!
**Urs Schreiber:**Deformation Theory of Algebras and Modules (March 7, 2011)- A conference on deformation theory.
**Tom Leinster:**Characterizing the Generalized Means (March 2, 2011)- A new characterization of generalized means?
**Urs Schreiber:**QVEST, Spring 2011 (March 1, 2011)- The second QVEST seminar meeting
**Tom Leinster:**Characterizing the*p*-Norms (March 1, 2011)- Exactly what makes the p-norms special?
**Urs Schreiber:**Differential Cohomology in a Cohesive Topos (February 28, 2011)- A text on general abstract and concrete differential geometry and cohomology.
**David Corfield:**Tarski’s Two Approaches to Modal Logic (February 25, 2011)- What links Tarski’s two approaches to modal logic?
**Urs Schreiber:**4d QFT for Khovanov Homology (February 23, 2011)- The identification of the 4d TQFT that has Khovanob homology as its observables.
**Urs Schreiber:**Structured Ring Spectra 2011 (February 18, 2011)- A conference on stable homotopy theory.
**Mike Shulman:**Concrete ∞-Categories (February 16, 2011)- Is there a notion of “concrete ∞-category”?
**John Baez:**Heron’s Formula (February 15, 2011)- See a 4-dimensional proof of Heron’s formula for the area of a triangle.
**Mike Shulman:**Punctual Local Connectedness (February 11, 2011)- A new paper by Peter Johnstone shows that Lawvere’s “pieces have points” axiom has some surprisingly strong consequences.
**John Baez:**Rényi Entropy and Free Energy (February 11, 2011)- Up to a temperature-dependent factor, Rényi entropy is just free energy!
**John Baez:**The Three-Fold Way (Part 5) (February 8, 2011)- Learn about the functors going between the categories of real, complex and quaternionic Hilbert spaces.
**Urs Schreiber:**Lurie: Higher Algebra (February 7, 2011)- Jacob Lurie has made his new book available.
**Urs Schreiber:**String-Math 2011 (February 4, 2011)- Announcement of the first _String-Math_ conference.
**Urs Schreiber:**Higher Structures in Topology and Geometry V (January 31, 2011)- Part V of the Goettingen/Hamburg Higher Structures meetings.
**John Baez:**The Three-Fold Way (Part 4) (January 29, 2011)- In the fourth episode of The Three-Fold Way, we’ll see how turning a particle around 180 degrees is related to making it go backwards in time.
**Guest:**Pictures of Modular Curves (XI) (January 27, 2011)- Working out how X0(N) is glued together
**David Corfield:**Coalgebraic Tangles (January 25, 2011)- I’m sinking in a sea of administrative duties at the moment, so for a bit of sanity I thought I’d jot down the glimmer of a thought I had. We spoke back here about the term model for a…
**John Baez:**Math Journal Workshop at MSRI (January 25, 2011)- There is a workshop on math journals at MSRI from February 14th to 16th, 2011.
**Mike Shulman:**Topos Theory Can Make You a Predicativist (January 23, 2011)- Working with higher topoi, and being used to negative thinking, can make predicative mathematics seem both more appealing and more feasible.
**David Corfield:**Seminar on Higher-Dimensional Algebra (January 23, 2011)- Some announcements
**Tom Leinster:**Category Theory 2011 (January 17, 2011)- The invited speakers for CT2011 have been announced.
**Guest:**Pictures of Modular Curves (X) (January 16, 2011)- Beginning to explore X0(N)
**Tom Leinster:**Another Editorial Board Resigns (January 11, 2011)- Not-so-new news about the Journal of Group Theory.
**Tom Leinster:**Magnitude of Metric Spaces: A Roundup (January 8, 2011)- Resources on magnitude of metric spaces.
**Urs Schreiber:**Stasheff:*Parallel Transport – Revisited*(January 7, 2011)- A note by Jim Stasheff on a homotopical notion of parallel transport.
**Guest:**Pictures of Modular Curves (IX) (January 7, 2011)- Smooth pictures of genus 0 and genus 1 curves X1(N)
**Simon Willerton:**Entailment and Implication (January 4, 2011)- Tell me the difference between these related ideas.
**David Corfield:**Impact! (January 3, 2011)- the impact of research
**Tom Leinster:**An Informal Introduction to Topos Theory (December 25, 2010)- A short paper on basic topos theory.
**Guest:**Pictures of Modular Curves (VIII) (December 23, 2010)- We continue modding out by translations, for N greater than 5
**Tom Leinster:**The Boyd Orr Centre, or: What is a Severed Horse Leg? (December 17, 2010)- Interdisciplinarity for mathematicians.
**Tom Leinster:**Poetic Refereeing (December 17, 2010)- Snappy excerpts from referees’ reports.
**John Baez:**Doodling in Math Class (December 17, 2010)- Vi Hart has some cool math videos.
**Urs Schreiber:**QVEST, Winter 2010 (December 16, 2010)- A seminar on aspects of higher category theory.
**John Baez:**A Global Glance on Categories in Logic (December 15, 2010)- Freely dowloadable papers on logic… for a limited time.
**David Corfield:**Purity of Method (December 14, 2010)- Purity of Method and Desargues’s theorem
**John Baez:**Higher Gauge Theory, TQFT and Quantum Gravity in Lisbon (December 14, 2010)- There’s a workshop and school on Higher Gauge Theory, TQFT and Quantum Gravity in Lisbon during February 7-13, 2011.
**Guest:**Pictures of Modular Curves (VII) (December 14, 2010)- We mod out modular curves by the translation subgroup
**John Baez:**The Three-Fold Way (Part 3) (December 13, 2010)- In the third episode of The Three-Fold Way, we’ll see that the “q” in “qubit” stands for
*quaternion*— at least when you think of them as representations of SU(2). **Tom Leinster:**Pri la Funkcia Ekvacio*f(x + y) = f(x) + f(y)*(December 10, 2010)- Fréchet, a functional equation, and a curious piece of history.
**John Baez:**The Library: Three Jeremiads (December 9, 2010)- Those of you interested in journal prices and the like will enjoy this article pointed out by David Roberts: Robert Darnton, The library: three jeremiads, The New York Review of Books, December, 2010. The author is a historian at Harvard…
**John Baez:**Quantum Foundations Mailing List (December 9, 2010)- Bob Coecke and Jamie Vicary have set up a mailing list on “quantum foundations”.
**Guest:**Pictures of Modular Curves (VI) (December 7, 2010)- The action of the projective group of units on tilings
**John Baez:**The Three-Fold Way (Part 2) (December 7, 2010)- The ‘three-fold way’ first manifests itself as a classification of unitary group representations — let’s see how that works.
**John Baez:**The Three-Fold Way (Part 1) (December 3, 2010)- In the first of a series of posts on real, complex and quaternionic quantum theory, we describe some problems with the real and complex theories.
**Urs Schreiber:**Hochschild Homology As Cohomology of Loop Space Objects (December 2, 2010)- Notes on Hochschild homology in terms of functions on higher loop space objects.
**John Baez:**Homological Algebra Puzzle (December 2, 2010)- A homological algebra puzzle about kernels and cokernels.
**John Baez:**Category Theory Puzzle (December 2, 2010)- A puzzle about categories enriched over the category of abelian groups.
**John Baez:**Solèr’s Theorem (December 1, 2010)- In 1995 Maria Pia Solèr proved an amazing theorem that characterizes Hilbert spaces over the real numbers, complex numbers and quaternions.
**Mike Shulman:**Internal Categories, Anafunctors and Localisations (November 29, 2010)- guest post by David Roberts This post is about my forthcoming paper, extracted from chapter 1 of my thesis: Internal categories, anafunctors and localisations and is also a bit of a call for examples from nn-category cafe visitors (see…
**John Baez:**State-Observable Duality (Part 3) (November 29, 2010)- The third and last of a short series of posts on the foundations of quantum theory: the Koecher–Vinberg classification of self-dual homogeneous convex cones, and its relation to observable-state duality.
**Tom Leinster:**What is Integral Geometry? (November 29, 2010)- A quick and dirty explanation.
**Guest:**Pictures of Modular Curves (V) (November 28, 2010)- How the rescaling group acts on modular curves
**John Baez:**State-Observable Duality (Part 2) (November 27, 2010)- The second of a short series of posts on the foundations of quantum theory: the theorem by Jordan, von Neumann and Wigner classifying ‘finite-dimensional formally real Jordan algebras’.
**David Corfield:**Over 4500 Entries in the nLab, and Its 2nd Birthday! (November 26, 2010)- Describing nLab on its second birthday
**John Baez:**State-Observable Duality (Part 1) (November 25, 2010)- The first of a short series of posts on the foundations of quantum theory: a review of normed division algebras.
**Mike Shulman:**Locally Constant Sheaves (November 25, 2010)- What is the right definition of a locally constant sheaf on a non-locally-connected space?
**Mike Shulman:**Universe Enlargement (November 23, 2010)- Ways to enlarge a category to live in a different universe.
**Guest:**Pictures of Modular Curves (IV) (November 22, 2010)- Using colour to understand the structure of tilings of modular curves.
**Guest:**Pictures of Modular Curves (III) (November 14, 2010)- Tilings of modular curves of level greater than 5, with non-positive curvature.
**Simon Willerton:**Integral Transforms and the Pull-Push Perspective, I (November 7, 2010)- Start to see how enriched profunctors can be viewed as categorifications of integral kernels.
**Urs Schreiber:**Cocycles for Differential Characteristic Classes (November 7, 2010)- Some aspects of Chern–Weil theory for higher bundles over higher Lie groups.
**Urs Schreiber:**Structures in a Cohesive ∞-Topos (November 6, 2010)- Inside any cohesive infinity-topos there exists a wealth of geometric and differential geometric structures.
**David Corfield:**Transforms (November 4, 2010)- What makes a transform tick?
**Guest:**Pictures of Modular Curves (II) (October 29, 2010)- Explaining some of the algebraic and geometrical background to the first article in the series.
**Mike Shulman:**Petit (∞,1)-Toposes (October 26, 2010)- (∞,1)-toposes, regarded as generalized spaces, provide a context in which to relate spaces to ∞-groupoids in a universal way.
**Tom Leinster:**A Short Warning (October 26, 2010)- On plagiarism.
**John Baez:**The Art of Math (October 22, 2010)- Sophie Hebden has written a nice gentle introduction to the work of John Baez’s students that was funded by a grant from the Foundational Questions Institute.
**Tom Leinster:**Benoît Mandelbrot (October 18, 2010)- Remarks on the Mandelbrot set and its wider context.
**Urs Schreiber:**Cohesive ∞-Toposes (October 14, 2010)- Some words on the notion of toposes of cohesion.
**John Baez:**Vanity and Ambition in Mathematics (October 14, 2010)- Some essays on mathematics by Jonah Sinick.
**Tom Leinster:**The Scottish Category Theory Seminar (October 13, 2010)- Announcement of the third meeting.
**Tom Leinster:**An Invitation (October 8, 2010)- Come in, come in!
**John Baez:**Pictures of Modular Curves (I) (October 6, 2010)- The first of a series of posts by Tim Silverman, with lots of pretty pictures.
**Mike Shulman:**Structure-Like Stuff (October 4, 2010)- The notion of ‘property-like structure’ suggests a categorified version of ‘structure-like stuff’ – are there any examples?
**John Baez:**A Hopf Algebra Structure on Hall Algebras (October 4, 2010)- Christopher Walker has a new paper, “A Hopf algebra structure on Hall algebras”.
**John Baez:**Democracy and Mathematics (October 4, 2010)- A message from André Joyal on reforming mathematics.
**John Baez:**Jacob Biamonte on Tensor Networks (September 29, 2010)- Jacob Biamonte wants your comments on his paper about string diagrams in quantum computation and condensed matter physics, which are called “tensor networks”.
**Tom Leinster:**Fetishizing*p*-Values (September 23, 2010)- The “cult of statistical significance”.
**Tom Leinster:**Bacard on Segal Enriched Categories (September 22, 2010)- Short description of a new paper on homotopy-enrichment
**Mike Shulman:**Grothendieck-Maltsiniotis ∞-categories (September 15, 2010)- Georges Maltsiniotis has posted a new definition of ∞-category, whose groupoidal version is said to be due to Grothendieck.
**David Corfield:**What Is This Category Enriched In? (September 14, 2010)- Searching for enrichment in a stochastic category
**John Baez:**Categories and Information in Oxford (September 11, 2010)- There’s a conference on “The categorical flow of information in quantum physics and linguistics” in Oxford from October 29th to 31st, 2010.
**John Baez:**Grothendieck’s “Tohoku” paper (September 7, 2010)- Michael Barr has made available an English translation of Grothendieck’s “Tohoku” paper.
**Tom Leinster:**Integral Geometry in Barcelona (September 6, 2010)- Short report from a meeting.
**Urs Schreiber:**Function*T*-Algebras on ∞-Stacks (September 1, 2010)- Herman Stel’s master thesis on Function Algebras on Infinity-Stacks.
**John Baez:**Bimonoids from Biproducts (September 1, 2010)- Every object in a category with finite biproducts is a bimonoid in a unique way.
**John Baez:**The Philosophy of the Logic of Sheaves (September 1, 2010)- On October 19 - 21, 2010, there’s a conference in Columbia on the philosophical aspects of sheaf logic.
**John Baez:**Categorification in Portugal (August 31, 2010)- The latest news from Jeffrey Morton.
**Simon Willerton:**Enriching Over a Category of Subsets (August 30, 2010)- Discover how enriched category theory leads to the definition of some generalized metrics on the space of continuous functions on the unit interval.
**Tom Leinster:**The Difference between Measure Zero and Empty Interior (August 28, 2010)- …is the difference between Lebesgue and Riemann integrability.
**John Baez:**This Week’s Finds (Week 301) (August 27, 2010)- The new
*This Week’s Finds*can be found over on Azimuth. **Mike Shulman:**The Geometry of Monoidal Fibrations? (August 26, 2010)- Does it exist?
**Tom Leinster:***Free High School Science Texts*Needs You! (August 25, 2010)- Proof-reading needed for open source textbooks.
**Tom Leinster:**What is the Langlands Programme? (August 21, 2010)- Please?
**Urs Schreiber:**Minicourse on*Nonabelian Differential Cohomology*(August 19, 2010)- A course on nonabelian differential cohomology.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 300) (August 11, 2010)- In “week300”, learn how to categorify the Riemann zeta function and get a groupoid whose cardinality is $\pi^2/6$.
**Tom Leinster:**Means (August 7, 2010)- Means of numbers and means of structures.
**John Baez:**Thermodynamics and Wick Rotation (August 6, 2010)- Help Mike Stay with some questions about the analogy between quantum mechanics and statistical mechanics!
**Mike Shulman:**LaTeX Overflow (August 4, 2010)- A new question+answer site for TeX questions
**Urs Schreiber:***Copenhagen Meeting on Differential Characters*(August 1, 2010)- A workshop on differential characters and Deligne cohomology.
**Tom Leinster:**Pullback-homomorphisms (August 1, 2010)- What does it signify for a map of algebras for a monad to be a pullback square?
**John Baez:**Azimuth (July 30, 2010)- John Baez has a new blog, called Azimuth, which is about ecological and technological issues.
**John Baez:**Zeta Functions: Dedekind Versus Hasse-Weil (July 28, 2010)- Some help on Hasse-Weil zeta functions, anyone?
**Mike Shulman:**Ternary Factorization Systems (July 24, 2010)- Defining ternary factorization systems in terms of binary ones.
**John Baez:**Commutative Separable Algebras II (July 22, 2010)- More questions about commutative separable algebras, especially over the integers.
**John Baez:**Commutative Separable Algebras (July 16, 2010)- What are some commutative semisimple algebras that aren’t separable?
**Mike Shulman:**What is a Theory? (July 14, 2010)- Is a syntactically presented theory as important as its walking model?
**Tom Leinster:**The Dold–Kan Theorem: Two Questions (July 12, 2010)- In Ab, many things become true.
**David Corfield:**Grinding to a Halt? (July 8, 2010)- future of the blog
**Urs Schreiber:***Day on Higher Category Theory*in Utrecht (July 5, 2010)- A workshop on higher category theory.
**John Baez:**Homological Algebra Puzzle (July 1, 2010)- What is the walking epimorphism?
**Alexander Hoffnung:**Categorification and Topological Invariants in Luminy (June 29, 2010)- We’re having a Workgroup on Categorification and Topological Invariants in Luminy.
**Urs Schreiber:**Dutch*Geometry and Quantum Theory*Conference 2010 (June 28, 2010)- Foruth annual conference of the Durch Geometriy And Quantum Theory Cluster.
**David Corfield:**Inevitability in Mathematics (June 28, 2010)- When is it unavoidable that a construct be found?
**Urs Schreiber:**Alan Carey’s 60th Birthday Conference (June 25, 2010)- This week at MPI Bonn is (or has been) taking place a conference in honor of Alan Carey’s 60th birthday. on “noncommutative geometry and index theory, statistical models, geometric issues in quantum field theory. Hamiltonian anomalies and bundle n-gerbes”….
**Mike Shulman:**Exact Squares (June 21, 2010)- The calculus of exact squares for computing with Kan extensions isn’t well-known in some circles, but it has the advantage of generalizing well to derivators and (∞,1)-categories.
**John Baez:**2010 Octoberfest (June 20, 2010)- There’s a category theory workshop on October 23-24, 2010 at Dalhousie University in Halifax, Canada.
**Simon Willerton:**Workshop on Music, Patterns and Mathematics (June 18, 2010)- A three-day meeting during August in England on Music, Patterns and Mathematics.
**Tom Leinster:**Algebraic Model Structures (June 16, 2010)- Algebraic model categories and weak factorization systems.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 299) (June 13, 2010)- In week299 of This Week’s Finds, hear about the school on Quantum Information and Computer Science that was recently held in Oxford, and the subsequent workshop on Quantum Physics and Logic. Then, find out what Bill Schmitt told me about combinatorics.
**John Baez:**Boycotting the Nature Publishing Group (June 9, 2010)- Nature Publishing Group is massively hiking its prices on many journals… and the University of California is considering ways to fight back.
**Urs Schreiber:**At “Geometry, Quantum Fields, and Strings: Categorial Aspects” (June 8, 2010)- An Oberwolfach workshop on higher categorical aspects of quantum field and string theory.
**David Corfield:**Vladimir Arnold, 12 June 1937 - 3 June 2010 (June 8, 2010)- Vladimir Arnold died last week
**Mike Shulman:**Tensor Categories in Fredericton (June 7, 2010)- Some highlights from the Tensor Categories session at the CMS 2010 summer meeting in Fredericton.
**John Baez:**Categories, Logic and Physics 7 at Birmingham (June 2, 2010)- The 7th Categories in Logic and Physics workshop will be in Birmingham, England on September 21st, 2010.
**John Baez:**The Quantum Whisky Club (May 28, 2010)- Fun at Oxford.
**David Corfield:**nLab Digest (May 27, 2010)- The latest and best at the wiki nLab
**Urs Schreiber:**Nonabelian Cohomology in Three (∞,1)-Toposes (May 26, 2010)- On nonabelian sheaf cohomology with constant coefficients in petit and gros (oo,1)-toposes.
**Mike Shulman:**Squeezing Higher Categories out of Lower Categories (May 18, 2010)- By using homotopy 2-categories and derivators, we can squeeze a lot of information out of (infinity,1)-categories using only comparatively easy 2-categorical machinery.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 298) (May 15, 2010)- In “week298” of This Week’s Finds, learn about finite subgroups of the unit quaternions, finite subloops of the unit octonions, Lie n-superalgebras built using division algebras, and Duff and Ferrara’s ideas connecting exceptional groups to quantum information theory. And learn what a "hyperdeterminant" is!
**David Corfield:**Quinn on Higher-Dimensional Algebra (May 10, 2010)- Frank Quinn’s critique of the tenth chapter of my book.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 297) (May 10, 2010)- In “week297” of This Week’s Finds, see knot sculptures and learn about special relativity in finance, lazulinos, some peculiar infinite sums, and a marvelous fact about the number 12. Then: Dirichlet forms and electrical circuits!
**Tom Leinster:**The Scottish Category Theory Seminar (May 8, 2010)- The second meeting of the Scottish Category Theory Seminar.
**David Corfield:**Back in Business (May 7, 2010)- The cafe is back again.
**Mike Shulman:**Understanding the Homotopy Coherent Nerve (April 29, 2010)- An easier-to-understand description of the left adjoint to the homotopy coherent nerve, due to Dugger and Spivak, enables us to make explicit computations of its hom-spaces, and better understand the relationship between quasicategories and simplicial categories.
**Mike Shulman:**Confessions of a Higher Category Theorist (April 29, 2010)- Why might we continue to search for more models for higher categories, aside from the simplicial ones that have proven so useful?
**John Baez:**This Week’s Finds in Mathematical Physics (Week 296) (April 26, 2010)- In week296 of This Week’s Finds, get a free copy of Jerry Shurman’s book
*The Geometry of the Quintic*, and read my attempt to construct a compact dagger-category whose morphisms are electrical circuits made of resistors. **Simon Willerton:**On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces (April 21, 2010)- See the details of a new paper on the magnitude of metric spaces.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 295) (April 17, 2010)- In “week295” of This Week’s Finds, learn about the principle of least power, Poincaré duality for electrical circuits — and a curious generalization of Hamiltonian mechanics that involves entropy as well as energy.
**David Corfield:**Paris in the Spring (April 15, 2010)- Mathematical ideas and how they combine
**Mike Shulman:**Symmetric Monoidal Bicategories and (n×k)-categories (April 8, 2010)- Monoidal bicategories can be constructed easily from “fibrant” monoidal double categories, and likewise for higher n-categories.
**Alexander Hoffnung:**New Paper on the Hecke Bicategory (April 6, 2010)- Take a look at a draft of the paper “The Hecke Bicategory”!
**Mike Shulman:**Extraordinary 2-Multicategories (March 30, 2010)- The still-hypothetical notion of “extraordinary 2-multicategory” generalizes a 2-category just enough to include extraordinary natural transformations.
**John Baez:**Alex Hoffnung Heading to Ottawa (March 25, 2010)- Alex Hoffnung got a job at the University of Ottawa!
**Simon Willerton:**Modeling Surface Diagrams (March 24, 2010)- Watch some videos to see how I’m trying to make 3d models of categorical surface diagrams.
**Mike Shulman:**Stack Semantics (March 23, 2010)- The stack semantics extends the internal logic of a topos to handle unbounded quantifiers in a category-theoretic way.
**Tom Leinster:**Normal Service is Resumed! (March 22, 2010)- Temporary software glitch is over.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 294) (March 22, 2010)- In “week294”, hear an account of Gelfand’s famous math seminar in Moscow. Read what Jan Willems thinks about control theory and bond graphs. Learn the proof of Tellegen’s theorem. Meet some categories where the morphisms are circuits, learn why a category object in Vect is just a 2-term chain complex.. and gaze at Saturn’s rings edge-on.
**Urs Schreiber:**We Have Moved (At Least Temporarily) (March 18, 2010)- The nCategory Cafe has moved to a different installation.
**John Baez:**An Invitation to Higher Gauge Theory (Again) (March 17, 2010)- You can now see ‘An invitation to higher gauge theory’ in near-final form.
**David Corfield:**Philosophy in the EMS Newsletter (March 16, 2010)- articles on realism in the EMS newletter
**Urs Schreiber:**Comment Bug (March 15, 2010)- The nCafé is currently affected by a bug that prevents comments to be posted. Here is a link to discussion on the nForum to alleviate this problem for the moment.
**John Baez:**Division Algebras and Supersymmetry II (March 13, 2010)- The real numbers, complex numbers, quaternions and octonions give Lie 2-superalgebras that describe the parallel transport of superstrings, and Lie 3-superalgebras that describe the parallel transport of 2-branes!
**Simon Willerton:**Intrinsic Volumes and Weyl’s Tube Formula (March 12, 2010)- Read about how the volume of a tube around a surface in 3-space depends only on intrinsic invariants of the surface.
**Urs Schreiber:**Séminaire Itinérant de Catégories (March 11, 2010)- A workshop on category theory and higher category theory.
**Urs Schreiber:**Summer School on Mathematical String Theory (March 9, 2010)- Summer school involving higher category theory in mathematical string theory.
**Tom Leinster:**A Perspective on Higher Category Theory (March 8, 2010)- Extended ramblings.
**David Corfield:**The Two Cultures Again (March 7, 2010)- Trying to make sense of the two cultures of mathematics
**John Baez:**Postdoc Position in Lisbon (March 5, 2010)- There’s a postdoc position in Lisbon for someone working on categorification or gerbes.
**John Baez:**Categorification in Faro (March 5, 2010)- The XIXth Oporto Meeting on Geometry, Topology and Physics will take place on July 19 – 23, 2010 in Faro, Portugal.
**John Baez:**2-Branes and Supergravity Theories (March 4, 2010)- What about supergravity theories in dimensions 4, 5, 7 and 11?
**Mike Shulman:**In Praise of Dependent Types (March 3, 2010)- Dependent types should be everyone’s friend!
**Urs Schreiber:**Algebraic Models for Higher Categories (March 2, 2010)- On (oo,1)-categories of algebraic oo-groupoids and algebraic (oo,1)-categories.
**Urs Schreiber:**Rational Homotopy Theory in an (oo,1)-Topos (March 2, 2010)- On rational homotopy theory and differential cohomology from the point of view of oo-topos theory.
**John Baez:**An Invitation to Higher Gauge Theory (March 1, 2010)- Read the draft of an expository paper on higher gauge theory by John Huerta and John Baez.
**Simon Willerton:**A Look at the Mathematical Origins of Western Musical Scales (February 26, 2010)- See how the rational numbers 2 and 3/2 gave birth to the Western musical scale.
**Urs Schreiber:**3000 and One Things to Think About (February 25, 2010)- Latest activity on the nLab, on occasion of entry number 3000.
**David Corfield:**(Infinity, 1)-logic (February 25, 2010)- In search of the internal logic of a $(\infty, 1)$-topos
**Urs Schreiber:**Higher Structures in Göttingen IV (February 15, 2010)- Announcement of the fourth workshop “Higher Structures in Göttingen”.
**John Baez:**Algorithmic Thermodynamics (February 14, 2010)- Read a new paper about the relation between thermodynamics and algorithmic information theory.
**Urs Schreiber:**Intrinsic Naturalness (February 12, 2010)- On naturalness and kludgyness, on concepts and tools in higher category theory.
**John Baez:**Grothendieck Said: “Stop” (February 10, 2010)- People were trying to republish Grothendieck’s
*magnum opus*. But then… **John Baez:**This Week’s Finds in Mathematical Physics (Week 293) (February 6, 2010)- In week293, catch up on recent papers and books about $n$-categories. Hear about last weekend’s Conference on the Mathematics of Environmental Sustainability and Green Technology at Harvey Mudd College. And learn how to think of networks of resistors as chain complexes which are also morphisms in a category.
**Tom Leinster:**Sheaves Do Not Belong to Algebraic Geometry (February 5, 2010)- They come out of thin categorical air.
**John Baez:**Quantum Physics and Logic at Oxford (February 3, 2010)- There will be a workshop on Quantum Physics and Logic on Saturday and Sunday May 29-30, 2010 at Oxford.
**Urs Schreiber:**Derived Synthetic Differential Geometry (February 2, 2010)- The geometry of structured infinity-toposes induced by starting from the pregeometry given by the site of smooth loci.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 292) (January 30, 2010)- In “week292”, learn about Henry Paynter’s “bond graphs” — a branch of applied category theory still waiting to be understood.
**Urs Schreiber:**6th Workshop on Categories, Logic and Foundations of Physics (January 25, 2010)- The 6th Workshop in a series organized by Bob Coecke and Andreas Döring.
**Mike Shulman:**Equivariant Stable Homotopy Theory (January 24, 2010)- Trying to understand equivariant stable homotopy theory from a higher-categorical perspective.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 291) (January 23, 2010)- In week291, listen to a crab canon on a Möbius strip, see how the Mandelbrot set mimics infinitely many Julia sets, and learn about 2-ports, 3-ports, and Poincaré duality for electrical circuits.
**David Corfield:**The Sacred and the Profane (January 18, 2010)- Contrasting two attitudes towards mathematics.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 290) (January 16, 2010)- In “week290” of This Week’s Finds, read about categorification in analysis. Ponder a number puzzle. Meet the five most popular 1-ports: resistances, capacitances, inertances, effort sources and flow sources. And learn a bit more rational homotopy theory!
**John Baez:**Going to Singapore (January 14, 2010)- John’s leave of absence was approved.
**Tom Leinster:***F*and the Shibboleth (January 14, 2010)- Thompson’s group F and free structured categories.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 289) (January 8, 2010)- In “week289” of This Week’s Finds, learn how E8 shows up in condensed matter physics. Continuing exploring the grand analogy between different physical systems! And learn about differential graded Lie algebras in rational homotopy theory.
**Urs Schreiber:**Quasicoherent ∞ -Stacks (January 8, 2010)- On the notion of quasicoherent sheaves and vector bundles in higher category theory.
**David Corfield:**Looking for Compatible Structure (January 8, 2010)- Can Fraisse limits support algebraic structure?
**Tom Leinster:**What You’re Doing is Good for You (January 6, 2010)- Open access articles get cited more.
**John Baez:**The Math of Environmental Sustainability at Harvey Mudd (January 6, 2010)- Ever wonder what mathematicians can do to help solve the environmental problems facing all of us? I do. Maybe this will help: 2010 Harvey Mudd College Mathematics Conference on the Mathematics of Environmental Sustainability and Green Technology, Harvey Mudd,…
**John Baez:**Coenergy (January 4, 2010)- What’s coenergy? Is ‘kinetic coenergy’ just kinetic energy regarded as a function of velocity rather than momentum?
**Mike Shulman:**Generalized Multicategories (January 4, 2010)- A brief introduction to generalized multicategories, in honor of a new draft of a paper about them.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 288) (January 1, 2010)- In “week288”, start exploring a huge set of analogies linking many branches of physics… and keep learning about rational homotopy theory. This time we’ll tackle differential graded Lie algebras.
**Alexander Hoffnung:**A Question About the Job Application Process (December 29, 2009)- A question on suggested deadlines and the meaning of “full consideration” in the job application process.
**Alexander Hoffnung:**Cobordism and Topological Field Theories Weeks 6 & 7 (December 21, 2009)- Weeks 6 and 7 of the cobordism and TFT seminar give a first look at $n$-dimensional TFT and the accompanying higher category theory!
**John Baez:**This Week’s Finds in Mathematical Physics (Week 287) (December 20, 2009)- In “week287” of This Week’s Finds, hear about the history of categorical logic, and continue learning about rational homotopy theory - especially Sullivan’s approach based on differential forms.
**David Corfield:**Galois Theory in Two Variables (December 18, 2009)- I have just returned from attending three days of the Final Workshop of the Newton Institute Non-Abelian Fundamental Groups in Arithmetic Geometry Programme. I was kindly invited by Café visitor Minhyong Kim, whose lecture we discussed a while ago. While…
**John Baez:**Can Five-Year-Olds Compute Coproducts? (December 13, 2009)- Some cognitive scientists are using category theory to understand the development of reasoning ability in children.
**Mike Shulman:**The Problem With Lax Functors (December 12, 2009)- There is no 3-category which includes 2-categories, lax functors, and any of the usual sorts of transformations.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 286) (December 9, 2009)- In week286 of This Week’s Finds, begin learning about rational homotopy! More will follow in Weeks to come…
**Mike Shulman:**Syntax, Semantics, and Structuralism II (December 8, 2009)- Structural set theories such as ETCS provide an alternate foundation for mathematics, which is arguably closer to mathematical practice than ZF-like “material” set theories.
**David Corfield:**When Naturality Fails (December 8, 2009)- Why should we make of important mathematical constructions which aren’t natural?
**Alexander Hoffnung:**Cobordism and Topological Field Theories Week 5 (December 7, 2009)- Week 5 of the cobordism and TFT seminar gives a short history of the interactions of quantum field theories and topology.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 285) (December 6, 2009)- In “week285” of This Week’s Finds, discover the beauty of roots. Then, hear what happened on the second day of the session on homotopy theory and higher algebraic structures in the AMS conference at U.C. Riverside!
**Alexander Hoffnung:**Cobordism and Topological Field Theories Week 4 (December 6, 2009)- Week 4 of the cobordism and TFT seminar introduces $2$-dimensional TFT’s and commutative Frobenius algebras.
**Mike Shulman:**Size, Yoneda, and Limits of Algebras (December 2, 2009)- A clever Yoneda argument shows that modulo size concerns, if completeness lifts to categories of algebras, then so do all individual limits. It’s interesting to compare how different approaches to size deal with this.
**John Baez:**Dangerous Knowledge (December 1, 2009)- There’s a BBC documentary called “Dangerous Knowledge”, about “four brilliant mathematicians — Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing — whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide”.
**Mike Shulman:**Feferman Set Theory (November 30, 2009)- (Strong) Feferman set theory provides a set-theoretic foundation for category theory that avoids many of the problems with other approaches such as Grothendieck universes.
**John Baez:**Dold–Kan Question (November 27, 2009)- Why is the normalized singular cochain complex of a space a differential graded coalgebra?
**David Corfield:**What’s So Special About the Rationals? (November 26, 2009)- Trying to link the rationals’ structures together.
**Simon Willerton:**Newton Fellowships at Sheffield (November 26, 2009)- Apply for a post-doc in Sheffield.
**John Baez:**The 50 Best Physics Blogs (November 26, 2009)- There’s a list of the 50 best physics blogs… and we’re on it!…
**Urs Schreiber:**Combinatorial Model Categories (November 25, 2009)- The theory of combinatorial model categories is currently scattered over the literature. Here is a collection of some links.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 284) (November 25, 2009)- In “week284” of This Week’s Finds, learn about the pentagon-decagon-hexagon identity, golden triangles, categorified quantum groups, the Halperin-Carlsson conjecture, real Johnson-Wilson theories, Picard 2-stacks, quasicategories, motivic cohomology theory, and toric varieties!
**Mike Shulman:**Equipments (November 23, 2009)- An enhanced structure on a 2-category, called a “proarrow equipment,” lets us define weighted limits and develop a good deal of “formal category theory.”
**David Corfield:**Mathematical Emotion (November 19, 2009)- Does mathematical writing express emotion?
**Alexander Hoffnung:**Cobordism and Topological Field Theories Week 3 (November 18, 2009)- Week 3 of the cobordism and TFT seminar introduces symmetric monoidal categories and TFT’s.
**Mike Shulman:**Combinatorial-Game Categories (November 16, 2009)- Joyal’s category of Conway games is the initial “combinatorial-game category.” What is the terminal one?
**David Corfield:**A Rose by Any Other Name (November 16, 2009)- Getting Ola Bratteli’s name right
**John Baez:**The 1000th Post on the n-Category Cafe (November 13, 2009)- Meet our new hosts: Alex Hoffnung, Tom Leinster, Mike Shulman and Simon Willerton!
**John Baez:**This Week’s Finds in Mathematical Physics (Week 283) (November 11, 2009)- See galaxies in visible, infrared and ultraviolet light. Read about who first discovered the regular icosahedron: Theatetus or the ancient Scots. And learn more about the geometry of the icosahedron!
**John Baez:**Courant Algebroids From Categorified Symplectic Geometry (November 10, 2009)- Read a draft of a new paper that gets Courant algebroids from 2-plectic manifolds.
**David Corfield:**Fraïssé Limits (November 6, 2009)- How should we understand a Fraïssé Limit category theoretically?
**John Baez:**Who Discovered the Icosahedron? (November 4, 2009)- Was it Theaetetus or the Scots?
**Urs Schreiber:**Notions of Space (November 4, 2009)- A survey of Jacob Lurie’s “Structured Spaces”.
**Tom Leinster:**An Adventure in Analysis (November 4, 2009)- How a problem about magnitude of metric spaces was solved.
**David Corfield:**The Arrow of Time in Cat (November 3, 2009)- Does the same broken symmetry that affects Set affect Cat?
**John Baez:**Interview with Manin (November 2, 2009)- Read a new interview with Yuri Manin.
**Mike Shulman:**Generalized Operads in Classical Algebraic Topology (October 30, 2009)- Two obscure-seeming kinds of generalized operad help make categorical sense out of two constructions in classical algebraic topology.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 282) (October 29, 2009)- In week282 of This Week’s Finds, visit Mercury: Learn how this planet’s powerful magnetic field interacts with the solar wind to produce flux transfer events and plasmoids. Then read about the web of connections between associative, commutative, Lie and…
**John Baez:**Math: Folk Wisdom in an Electronic Age (October 26, 2009)- People are starting to debate the best ways to use the internet to gather, store and access ‘folk wisdom’ about mathematics.
**Urs Schreiber:**Mathematical Foundations of Quantum Field and Perturbative String Theory (October 24, 2009)- A Call for Comments on a Call for Papers on an issue on Mathematical Foundations of Quantum Field and Perturbative String Theory.
**John Baez:**Cheaper Online Textbooks? (October 22, 2009)- The U.S. Department of Education is spending $300,000 to develop online textbooks. But how about free online textbooks?
**Alexander Hoffnung:**Cobordism and Topological Field Theories Week 2 (October 22, 2009)- Week 2 of the cobordism and TFT seminar introduces oriented manifold and cobordisms in a more formal way.
**Mike Shulman:**Aesthetics of Commutative Diagrams (October 21, 2009)- How to lay out a large commutative diagram aesthetically?
**John Baez:**This Week’s Finds in Mathematical Physics (Week 281) (October 20, 2009)- In “week281” of This Week’s Finds, learn about the newly discovered ring of Saturn, tilings with 5-fold and 10-fold quasisymmetry, and the latest news on quantum gravity from Corfu.
**Mike Shulman:**Syntax, Semantics, and Structuralism, I (October 19, 2009)- A brief summary of syntax and semantics of type theory and logic, as a basis for continuing discussions about structural and material foundations.
**Simon Willerton:**Halmos on Writing Mathematics (October 19, 2009)- Over in a discussion at Math Overflow I was reminded about Halmos’ great article on writing mathematics, which I highly recommend to all graduate students (or anyone else, for that matter). P. R. Halmos, How to write mathematics, L’Enseignement…
**David Corfield:**Math Overflow (October 19, 2009)- The new math question and answer site
**John Baez:**Girih by Egan (October 16, 2009)- If you like quasicrystals, check out this new applet by Greg Egan.
**Alexander Hoffnung:**Seminar on Cobordism and Topological Field Theories at UCR (October 15, 2009)- Lecture 1 of the Seminar on Cobordism and Topological Field Theories at UCR.
**David Corfield:**A Dual for Set? (October 14, 2009)- Question about structural set theory
**Urs Schreiber:**Path-Structured Smooth (∞,1)-Toposes (October 13, 2009)- A talk on a context for higher synthetic differential geometry.
**Alexander Hoffnung:**MSRI Workshops on Knot Homology Theories (October 11, 2009)- Announcement of MSRI workshops on knot homology theories.
**Simon Willerton:**More Magnitude of Metric Spaces and Problems with Penguins (October 10, 2009)- Learn about the tenuous link between emperor penguins and the magnitude of metric spaces.
**Mike Shulman:**Associativity Data in an (∞,1)-Category (October 8, 2009)- Guest post by Emily Riehl A popular slogan is that (∞,1)(\infty,1)-categories (also called quasi-categories or ∞\infty-categories) sit somewhere between categories and spaces, combining some of the features of both. The analogy with spaces is fairly clear, at least to…
**John Baez:**Borisov on Shapes of Cells in*n*-Categories (October 6, 2009)- Can you comment on a paper about comparing definitions of $n$-categories?
**John Baez:**Israel Gelfand, September 2, 1913 - October 5, 2009 (October 6, 2009)- Israel Gelfand died on Monday October 5th, at the age of 96.
**David Corfield:**Shell What? (October 5, 2009)- My ol blog pages
**Alexander Hoffnung:**Special Issue on “Categorification in Representation Theory” (October 2, 2009)- Announcement of Special Issue on “Categorification in Representation Theory”
**David Corfield:**Steve Lack Wins Medal (October 2, 2009)- Steve Lack wins Australian Mathematics Society medal.
**John Baez:**Sex Advice From Mathematicians (October 1, 2009)- The most amusing email of the day.
**Urs Schreiber:**D’Auria-Fré-Formulation of Supergravity (September 30, 2009)- An nLab entry on the D’Auria-Fre formulation of supergravity and its interpretation as a higher gauge theory.
**John Baez:**What Do Mathematicians Need to Know About Blogging? II (September 30, 2009)- Help me write an opinion piece on math blogging for the
*AMS Notices*! **John Baez:**This Week’s Finds in Mathematical Physics (Week 280) (September 28, 2009)- Hear about the courses Ashtekar and Rovelli taught at the quantum gravity summer school at Corfu.
**David Corfield:**The Mathematical Vocation (September 28, 2009)- Collingwood on the value of the pure mathematician.
**John Baez:**Questions on*n*-Curvature (September 26, 2009)- Some questions for Urs.
**John Baez:**Homotopy Theory and Higher Algebraic Structures at UC Riverside (September 24, 2009)- We’re having a special session on homotopy theory and higher algebraic structures here at UCR on November 7th and 8th. I hope you can come by!
**John Baez:**Movie Trivia Question (September 24, 2009)- A trivia question involving Star Trek and two Bogart movies.
**Urs Schreiber:**A Seminar on a Geometric Model for TMF (September 21, 2009)- A seminar on geometric models for tmf cohomology theory by Stephan Stolz and Peter Teichner.
**David Corfield:**Proof by Coinduction (September 18, 2009)- How similar is coinduction to induction?
**Urs Schreiber:**A Seminar on Gromov-Witten Invariants (September 17, 2009)- A mathematician’s seminar on Gromov-Witten theory.
**Urs Schreiber:***n*Lab - Latest Changes (September 17, 2009)- Trying to install better software support for logging latest changes to the nLab.
**Urs Schreiber:**Inbound Citations (September 15, 2009)- A list of citations of entries at the nCafé blog and/or at the nLab wiki.
**John Baez:**Towards a Computer-Aided System for Real Mathematics (September 10, 2009)- Arnold Neumaier is seeking useful comments on his proposal for a Formal Mathematical Language for doing and communicating mathematics on the computer.
**Urs Schreiber:**Interactive Science Symposium (September 8, 2009)- A sympositum on the use of web technology for communication among scientists.
**John Baez:**The Pi Calculus II (September 8, 2009)- Mike Stay tries to explain the pi calculus to John Baez.
**David Corfield:**Coalgebraic Modal Logic (September 7, 2009)- Putting together ideas of modal logic and coalgebra with ways of modelling first-order theories.
**Urs Schreiber:**A Seminar on a Survey of Elliptic Cohomology (September 6, 2009)- An nLab entry accompanying a seminar on elliptic cohomology.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 279) (September 6, 2009)- In week279, read about division algebras and supersymmetry - and a bit about tuning systems and infinitely groupoidified logic.
**Urs Schreiber:**Traces in Bicategories (September 4, 2009)- An article by Ponto and Shulman on traces in bicategories.
**David Corfield:**Where Have All the Solitons Gone? (September 2, 2009)- A request for examples of applied soliton theory.
**John Baez:**Division Algebras and Supersymmetry (September 1, 2009)- Learn how special properties of the reals, complexes, quaternions and octonions make supersymmetry tick in dimensions 3, 4, 6, and 10.
**Urs Schreiber:***n*Lab Migration Done (September 1, 2009)- The
*n*Lab wiki has successfully migrated to a better server. **John Baez:**SSE Composite Index Bubble II (September 1, 2009)- Was Didier Sornette right in predicting a plunge in the Shanghai Stock Exchange (SSE) Composite Index?
**Alexander Hoffnung:**Ben-Zvi’s Lectures on Topological Field Theory III (September 1, 2009)- This is the third set of notes on David Ben-Zvi’s lectures on topological field theory at Northwestern University.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 278) (August 30, 2009)- In week278, hear the latest about Betelgeuse. Read how red supergiants spew out dust which eventually forms planets like ours. Watch a hypervelocity collision in a distant solar system. Learn the new way to make graphene, and read my history of the Earth - for physicists. And when you’re ready: dive into groupoidification!
**David Corfield:**Feynman the Babylonian (August 27, 2009)- Feynman’s concerns about an axiomatic approach
**John Baez:**Higher-Dimensional Algebra VII: Groupoidification (August 23, 2009)- Check out our new paper “Higher-dimensional algebra VII: groupoidification”.
**Urs Schreiber:**Notes on Differential Nonabelian Cohomology (August 21, 2009)- Notes on differential nonabelian cohomology.
**Urs Schreiber:***n*Lab Migration (August 19, 2009)- Request for help with testing the migrated nLab wiki installation.
**David Corfield:**Chasing around the Triangle (August 17, 2009)- Steve Awodey compares foundational frameworks: sets, types and categories.
**John Baez:**The Pi Calculus (August 15, 2009)- The pi calculus is a process calculus. But what does
*that*mean? **John Baez:**Asymptotics of the Magnitude of Metric Spaces (August 13, 2009)- Learn about Tom Leinster and Simon Willerton’s new paper ‘On the asymptotic magnitude of subsets of Euclidean space’.
**John Baez:**Dan Ghica on Software vs Hardware (August 13, 2009)- A few notes on Dan Ghica’s ideas about ‘function interface models for hardware compilation’ — a potentially very practical application of symmetric monoidal closed categories.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 277) (August 12, 2009)- In “week277”, find out what’s a million times thinner than paper, stronger than diamond, a better conductor than copper, and absorbs exactly pi times the fine structure constant of the light you shine through it.
**Urs Schreiber:**Question about ∞-Colimits (August 10, 2009)- What can one say about hocolimits over domains that are themselves hocolimits?
**John Baez:**What Do Mathematicians Need to Know About Blogging? (August 9, 2009)- You can help me write an opinion piece on math blogging for the AMS
*Notices*. **John Baez:**Computation and the Periodic Table II (August 8, 2009)- I’m giving a talk at LICS 2009 – this year’s annual conference on Logic in Computer Science.
**Urs Schreiber:**Question on Synthetic Differential Forms (August 3, 2009)- On a realization of differential forms as cosimplicial generalized smooth algebras.
**John Baez:**Groupoidfest 09 (August 2, 2009)- Groupoidfest 09 is being held on October 24–25, 2009 at the Department of Mathematics of the University of Colorado, Boulder, and it’s being organized by Arlan Ramsay.
**John Baez:**Question About Exotic Smooth Structures (July 30, 2009)- I’ve heard an interesting story about using Khovanov homology to help prove the existence of exotic smooth structures in 4 dimensions. Could you help check to see if it’s correct?
**Urs Schreiber:**Question on Models for (∞,1)-Functor Categories (July 29, 2009)- How and under which conditions are (∞,1)–functor categories modeled my simplicial model structures on functor categories.
**John Baez:**Generalizing “One-to-One” and “Onto” (July 28, 2009)- Jonas Frey has settled some questions about the 2-categorical duals of the notions of ‘full’ and ‘fully faithful’.
**John Baez:**Arithmetic Geometry at the Newton Institute (July 24, 2009)- There will be an introductory workshop on non-abelian fundamental groups in arithmetic geometry at the Newton Institute from July 27th to 31st, 2009.
**Urs Schreiber:**Verity on Descent for Strict ω-Groupoid Valued Presheaves (July 23, 2009)- Dominc Verity characterizes the descent condition for infinity-groupoid valued presheaves that happen to take values in strict infinity-groupoids.
**Urs Schreiber:**nLab – How to get started (July 23, 2009)- Quick easy information on how to make use of the nLab.n
**John Baez:**Thomas Noll’s Talks at Chicago on Mathematical Music Theory (July 23, 2009)- Thomas Fiore has written a guest post about Thomas Noll’s talks on mathematical music theory.
**John Baez:**SSE Composite Index Bubble? (July 21, 2009)- The geophysicist Didier Sornette and his coauthors predict that a bubble in the SSE will burst between July 17th and July 27th, with 60% probability.
**John Baez:**The Monads Hurt My Head — But Not Anymore (July 19, 2009)- I’m ready to learn some more advanced topics in the theory of monads. Here’s a list of them. Can you help me out?
**David Corfield:**Being Tentative on*n*Lab (July 17, 2009)- Is nLab the best place to discuss tentative ideas?
**John Baez:**A Puzzle From Gavin Wraith (July 16, 2009)- A puzzle by Gavin Wraith, who found a wonderfully high-powered solution.
**John Baez:**Searching for a Video Proof of “Seven Trees in One” (July 16, 2009)- James Propp is searching for a video proof that he saw once.
**John Baez:**Ben Zvi’s Lectures on Topological Field Theory II (July 14, 2009)- This is the second set of notes on David Ben-Zvi’s lectures on topological field theory at Northwestern University.
**John Baez:**Categorifying Nicomachus of Gerasa’s Equation (July 13, 2009)- Can you find a bijective proof of a formula for the sum of cubes of the first $n$ natural numbers?
**John Baez:**Ben-Zvi’s Lectures on Topological Field Theory I (July 8, 2009)- These are the first of some notes of David Ben-Zvi’s lectures at a workshop on topological field theories at Northwestern University, held in May 2009.
**John Baez:**A Prehistory of n-Categorical Physics II (July 7, 2009)- I’m back to writing that Prehistory of
*n*-Categorical Physics. I want to finish it before I die. **David Corfield:**Generalized Homotopy Theory (July 6, 2009)- On doing the parallel generalization on homotopy as has been done on cohomology.
**John Baez:**Course on Topological Quantum Field Theory in Almería (July 5, 2009)- There will be a school on TQFTs in Almería, Spain from October 19th to 23rd, 2009.
**John Baez:**Open Access to Taxpayer-Funded Research (July 2, 2009)- nn-Café regulars will know about Representative Conyer’s bill that would repeal the National Institute of Health’s public access policy and forbid other US funding agencies from mandating open access to research papers written with the help of federal grant…
**John Baez:**Elsevier Pays for Favorable Book Reviews (July 2, 2009)- Elsevier offered gift certificates to academics to write 5-star reviews of a book on Amazon and Barnes & Noble.
**Urs Schreiber:**Laubinger on Lie Algebras for Frölicher Groups (July 1, 2009)- Martin Laubinger studies Lie algebras of groups with generalized smooth manifold structure: Froelicher groups.
**John Baez:**Springer Verlag Publishes ‘Proof’ of Goldbach’s Conjecture (June 30, 2009)- Springer Verlag has published a book purporting to give elementary proofs of Fermat’s Last Theorem and Goldbach’s Conjecture.
**John Baez:**This Book Needs a Title (June 26, 2009)- Peter May and I are almost done editing a volume of survey articles that should help people get going on higher categories. What should we call it?
**David Corfield:**Cohomology and Homotopy (June 26, 2009)- On the duality between cohomology and homotopy
**John Baez:**Hopf Algebras in Luxembourg (June 20, 2009)- On July 13th to 17th 2009 there will be a conference and school on Hopf algebras in Luxembourg.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 276) (June 20, 2009)- In “week276”, read about Betelgeuse, the Local Bubble, the Loop I Bubble, the cloudlets from Sco-Gen, and the “local fluff”. Get to know the nLab. And learn how Paul-André Mélliès and Nicolas Tabareau have taken some classic results of Lawvere on algebraic theories and generalized them to other kinds of theories, like PROPs.
**David Corfield:**Kan Lifts (June 20, 2009)- Where is the dual of thenotion of Kan extensions?
**David Corfield:**Accessible Even to a Philosopher (June 16, 2009)- Frenkel’s new expository paper
**John Baez:**2-Branes in 11 Dimensions (June 15, 2009)- A question about 2-branes in 11 dimensions.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 275) (June 14, 2009)- Read about progress on proving the Cobordism Hypothesis.
**John Baez:**Final Exams Again (June 10, 2009)- What’s the funniest thing you’ve seen on a final exam this year?
**Urs Schreiber:**Strings, Fields, Topology in Oberwolfach (June 8, 2009)- Notes and material on Oberwolfach workshop on Strings, Fields and Topology
**John Baez:**Algebraic Geometry for Category Theorists (June 8, 2009)- You can now see the introduction to a paper James Dolan is writing about algebraic geometry for category theorists. You can also see 5 lectures he gave on this topic.
**John Baez:**Nonabelian Algebraic Topology (June 6, 2009)- Ronald Brown, Philip J. Higgins and Rafael Sivera have come out with a book called
*Nonabelian algebraic topology: homotopy groupoids and filtered spaces*. You can download it for free. **David Corfield:**The Elusive Proteus (June 5, 2009)- Continuing the thought that mathematics and physics share in having difficult to articulate general principles.
**David Corfield:**Mathematical Principles (June 3, 2009)- Cassirer describes certain physical principles. Are there corresponding ones in mathematics?
**John Baez:**Journal Club — Geometric Infinity-Function Theory — Week 6 (June 3, 2009)- Alex Hoffnung on topological field theories.
**John Baez:**Treq Lila (June 2, 2009)- I made an album
*Treq Lila*, in which all the melodies were created by cellular automata. **John Baez:**The Mathematics of Music at Chicago (May 31, 2009)- Thomas Noll will give some talks on the mathematics of music at the University of Chicago.
**John Baez:**Quantum Gravity and Quantum Geometry in Corfu (May 28, 2009)- There will be a summer school in Corfu from September 13th to September 20th, with courses on quantum gravity, renormalization and higher gauge theory.
**David Corfield:**Metric Coinduction (May 28, 2009)- When does coinduction differ from induction?
**Urs Schreiber:**Alm on Quantization as a Kan Extension (May 26, 2009)- An observation on a relation between Kan extensions and path integral quantization – by Johan Alm.
**John Baez:**Elsevier Journal Prices (May 24, 2009)- Can you help Ted Jacobson and I find information about the price of Elsevier journals compared to journals put out by other publishers?
**John Baez:**Charles Wells’ Blog (May 22, 2009)- Charles Wells has a new blog. I especially like the entries on ‘sketches’ and on ‘how “math is logic” ruined mathematics for a generation’.
**David Corfield:**Where is the Philosophy of Physics? (May 19, 2009)- Should we devote more time to the philosophy of physics?
**Urs Schreiber:**TFT at Northwestern (May 19, 2009)- Reports from a conference on extended topological quantum field theory at Northwestern University.
**John Baez:**A Prehistory of*n*-Categorical Physics (May 18, 2009)- Help John Baez write a paper on the prehistory of
*n*-categorical physics — the prehistory of a history that hasn’t quite started yet. **John Baez:**Higher Structures in Göttingen III (May 18, 2009)- There’s another workshop on higher structures at the Courant Research Centre in Göttingen. It sounds great!
**Urs Schreiber:**Journal Club – Geometric Infinity-Function Theory – Week 4 (May 18, 2009)- Chris Brav reviews technical details about tensor products and integral transforms of quasi-coherent sheaves on perfect stacks.
**David Corfield:**The Relevance of Predicativity (May 15, 2009)- Whether the predicative/impredicative split is noticeable in mainstream mathematics.
**Urs Schreiber:**Journal Club – Geometric Infinity-Function Theory – Week 3 (May 11, 2009)- This week in our Journal Club on [[geometric ∞\infty-function theory]] Bruce Bartlett talks about section 3 of “Integral Transforms”: perfect stacks. So far we had Week 1: Alex Hoffnung on Introduction Week 2, myself on Preliminaries See here for…
**John Baez:**Smooth Structures in Ottawa II (May 9, 2009)- A summary of some talks at the Fields Workshop on Smooth Structures in Logic, Category Theory and Physics.
**David Corfield:**In Search of Terminal Coalgebras (May 8, 2009)- Trying to understand a talk on terminal coalgebras
**John Baez:**Odd Currency Puzzle (May 7, 2009)- A puzzle concerning odd American currency.
**David Corfield:***n*Lab - More General Discussion (May 6, 2009)- Discussing nLab
**David Corfield:**Gerundives (May 5, 2009)- Naming ‘things to be categorified’
**Urs Schreiber:**Journal Club – Geometric Infinity-Function Theory – Week 2 (May 5, 2009)- Preliminaries for the discussion of geometric infinity-function theory: higher categories, higher sheaves, higher algebra, higher traces and what it all means.
**John Baez:**The Foibles of Science Publishing (May 4, 2009)- News about Elsevier’s fake medical journal and problems at
*Scientific American*. **David Corfield:**Taming the Boundless (April 30, 2009)- How in the modern period the boundless is divided between the infinite and the indefinite, and the former approached mathematically.
**John Baez:**A Riddle (April 29, 2009)- See if you can solve this riddle.
**John Baez:**The Earth - For Physicists (April 28, 2009)- A history of the Earth in 3000 words or less, for physicists.
**Urs Schreiber:**Journal Club – Geometric Infinity-Function Theory – Week 1 (April 27, 2009)- Journal Club on Geometric oo-Function theory part I: Introduction to integral transforms.
**John Baez:**‘Kervaire Invariant One Problem’ Solved (April 23, 2009)- It seems Michael Hopkins has solved a famous old topology problem due to Milnor and Kervaire.
**David Corfield:**Afternoon Fishing (April 23, 2009)- A question on a picture of Spec(Q).
**John Baez:**Comparative Smootheology, IV (April 23, 2009)- Take a look at Martin Laubinger’s thesis on smooth spaces.
**Urs Schreiber:**Master in Mathematical Physics at Hamburg University (April 20, 2009)- A new master program in mathematical physics at University of Hamburg.
**John Baez:**Kamnitzer on Categorifying Tangle Invariants (April 14, 2009)- A brief summary of a talk by Joel Kamnitzer at the workshop on Categorification and Geometrisation from Representation Theory.
**Urs Schreiber:**Report on 88th Peripatetic Seminar on Sheaves and Logic (April 11, 2009)- Mike Shulman reports on the 88th Peripatetic Seminar on Sheaves and Logic
**Urs Schreiber:**Journal Club – Geometric Infinity-Function Theory (April 10, 2009)- A place to discuss and learn about the work by Ben-Zvi/Francis/Nadler on geometric infinity-function theory and its application in infinity-quantum field theory.
**John Baez:**Commutative Diagrams in Toulouse (April 8, 2009)- There will be a workshop on Computer Algebra Methods and Commutativity of Algebraic Diagrams in Toulouse on October 16th and 17th, 2009.
**Urs Schreiber:**Graphical Category Theory Demonstrations (April 7, 2009)- On computer aided diagrammatic reasoning.
**John Baez:**Categorification and Topology (April 7, 2009)- Here are the slides for a talk explaining some hypotheses relating
*n*-categories and topology, and Jacob Lurie’s new work on these hypotheses. **Urs Schreiber:**Cohomology (April 4, 2009)- On the general (infinity,1)-categorical notion of cohomology.
**David Corfield:**A River and a Trickle (April 3, 2009)- On the brief appearance of Lakatos in a Handbook of philosophy of mathematics
**David Corfield:**Convex Spaces (April 1, 2009)- Fritz on convex spaces as algebras for the finite Giry monad
**John Baez:**Categories, Quanta and Concepts at the Perimeter Institute (March 31, 2009)- There will be a workshop called Categories, Quanta, Concepts at the Perimeter Institute from June 1st to 5th, 2009.
**David Corfield:**First-Order Logical Duality (March 27, 2009)- Announcing a recent PhD thesis which looks to extend Stone duality to a duality between syntax and semantic for first-order theories.
**John Baez:**The Algebra of Grand Unified Theories III (March 25, 2009)- Is the SO(10) grand unified theory of particle physics still experimentally viable? A plea for help from phenomenologists.
**David Corfield:**The Borromean Link Configuration (March 23, 2009)- Looking for a general formulation of situations where 3 entities are pairwise unconnected, yet connected as a triple.
**John Baez:**Unitary Representations of the Poincaré Group (March 22, 2009)- This blog entry is supposed to be a forum for learning and discussing Wigner’s classification of the representations of the Poincaré group. Ask and answer questions about this subject here!
**John Baez:**Open Access at MIT (March 22, 2009)- MIT has passed an open-access mandate!
**Urs Schreiber:**Twisted Differential String- and Fivebrane-Structures (March 20, 2009)- An article on twisted differential nonabelian cohomology and its application to anomaly cancellation in string theory.
**John Baez:**The Algebra of Grand Unified Theories II (March 16, 2009)- If you know a bit of group representation theory and you’ve always wanted to understand some particle physics, now is your chance: read a gentle expository account of the algebraic patterns lurking behind three famous Grand Unified Theories!
**David Corfield:**Geometric Help Wanted (March 13, 2009)- What does a result of Felix Klein mean?
**John Baez:**The Okubo Algebra (March 13, 2009)- Okubo invented a mysterious 8-dimensional division algebra while pondering quarks. What’s it all about?
**John Baez:**This Week’s Finds in Mathematical Physics (Week 274) (March 12, 2009)- Learn about infinite-dimensional 2-vector spaces and representations of 2-groups on these.
**David Corfield:**Synergism (March 10, 2009)- Are we maximising the collaborative power of our blog?
**John Baez:**Banning Open Access II (March 8, 2009)- Conyers defends his bill which would prohibit the National Institute of Health from mandating open access to the research they fund.
**John Baez:**Super-Yang-Mills Theory (March 5, 2009)- Why is supersymmetic Yang–Mills theory really supersymmetric? A plea for help.
**Urs Schreiber:**Question on Homotopical Structure on SimpSet (March 5, 2009)- A question on (directed) path objects in Kan complexes and weak Kan complexes.
**David Corfield:**The Stabilizer of a Subcategory (March 2, 2009)- Extending John Baez’s definition of the stabilizer of an object in a category to general subcategories
**David Corfield:**Two Streams in the Philosophy of Mathematics (March 2, 2009)- Announcing a philosophy of mathematics conference
**Urs Schreiber:**Question on Geometric Function Theory (February 27, 2009)- On the choice of “nice generalized geometric functions” in a general context of oo-stacks.
**John Baez:**New Structures for Physics III (February 27, 2009)- Mike Stay and I have finished what we hope is the final version of our paper for Bob Coecke’s book on New Structures for Physics. Peter Selinger’s paper for this book is also done.
**David Corfield:**Lakatos as Dialectical Realist (February 26, 2009)- Considering Lakatos’s idea of mathematicians aiming at the real through dialogue
**John Baez:**Monoidal Closed Categories And Their Deviant Relatives (February 22, 2009)- I’m curious to know whether a deviant definition of ‘monoidal closed category’ is equivalent to the usual one, or see counterexamples if it’s not.
**John Baez:**Homotopical Algebraic Geometry in Seville (February 21, 2009)- There will be a school on Homotopy Theory and Algebraic Geometry in Seville from September 7th to 13th, 2009.
**John Baez:**Satanist Rocketeer Puzzle (February 19, 2009)- Two puzzles.
**David Corfield:**Can -oids save Group Theory 101? (February 17, 2009)- Intriguing questions from Lieven le Bruyn.
**David Corfield:**Dialectical Realism in Utrecht (February 16, 2009)- My talk on Lautman in Utrecht
**John Baez:**Banning Open Access (February 16, 2009)- There’s a bill in the US House of Representatives that would repeal the NIH’s public access policy… and ban similar policies by other federal agencies.
**John Baez:**Baković and Jurčo on Classifying Topoi for Topological Bicategories (February 14, 2009)- Igor Bakoviċ and Branislaw Jurčo have a new paper out.
**John Baez:**Lurie on TQFT and the Cobordism Hypothesis (February 13, 2009)- Watch videos of four talks by Jacob Lurie at the University of Texas at Austin.
**John Baez:**The Algebra of Grand Unified Theories (February 10, 2009)- John Huerta is taking his oral exam soon. He’ll give a talk on the group representation theory underlying three famous grand unified theories, and how they fit together in a larger pattern. You can see the slides now.
**John Baez:**Link Homology in Paris (February 10, 2009)- In 2009 there will be a summer school on link homology in Paris.
**Urs Schreiber:**Moerdijk on Infinity-Operads (February 6, 2009)- Ieke Moerdijk on (infinity,1)-operads.
**Urs Schreiber:**Dendroidal Sets and Infinity-Operads (February 5, 2009)- Notes from a talk by Ieke Moerdijk on dendroidal sets, with a few remarks on presheaves on the category of posets.
**John Baez:**The Cocktail Party Version (February 4, 2009)- Jeffrey Morton explains what he’s working on.
**David Corfield:**Last Person Standing (February 2, 2009)- On the effects of nobody being left who understands the totality of the huge proof of a result.
**John Baez:**New York City Category Seminar (January 29, 2009)- There’s a new seminar on categories, organized by Noson Yanofsky, meeting regularly at the CUNY Grad Center in Manhattan.
**John Baez:**The Third Time is the Charm (January 29, 2009)- A puzzle about math and physics.
**John Baez:**Can Particle Physicists Regain Control of Their Journals? (January 28, 2009)- Learn about the Sponsoring Consortium for Open Access Publishing in Particle Physics.
**John Baez:**Bruce Bartlett’s Thesis (January 27, 2009)- Bruce Bartlett’s thesis on topological quantum field theory is on the arXiv now.
**John Baez:**Categorified Symplectic Geometry and the String Lie 2-Algebra (January 26, 2009)- A new paper shows how to build the string Lie 2-algebra by taking a compact Lie group with its canonical closed 3-form and then using ideas from multisymplectic geometry.
**David Corfield:**Abstract Stone Duality (January 26, 2009)- Paul Taylor’s programme to provide a computable foundation for topology
**John Baez:**Limits in the 2-Category of 2-Hilbert Spaces (January 22, 2009)- What limits are present in the 2-category of finite-dimensional 2-Hilbert spaces?
**John Baez:**Petit Topos, Gros Topos (January 22, 2009)- What’s the relation between the petit topos and the gros topos of a scheme?
**John Baez:**Hopf Algebras from Posets (January 14, 2009)- From a sufficiently nice category, we can get a Hopf algebra. But what, exactly, does ‘sufficiently nice’ mean? Maybe you can help me out.
**David Corfield:**The Space of Robustness (January 14, 2009)- A discussion of a blog post by Tim Gowers on the structure of mathematics.
**David Corfield:**Truth as Value and Duty (January 12, 2009)- Manin on mathematical wisdom
**Urs Schreiber:**Ben-Zvi on Geometric Function Theory (January 8, 2009)- An exposition by David Ben-Zvi on central ideas of geometric function theory, its categorification and relation to groupoidification.
**David Corfield:**Categories, Logic and Physics in London (January 8, 2009)- Reporting on the 4th CLP workshop
**Urs Schreiber:***n*Lab – General Discussion (January 4, 2009)- A place for general discussion concerning the nLab.
**John Baez:**The Toric Variety Associated to the Weyl Chambers (December 31, 2008)- The Weyl chambers of a simple Lie algebra form a ‘fan’, and this fan gives a toric variety. But what’s that toric variety like? What does it mean?
**David Corfield:**Organizing the Pages at*n*Lab (December 31, 2008)- How to organise pages at nLab
**John Baez:**Joint Math Meetings in Washington DC (December 31, 2008)- I’m going to the Joint Mathematics Meetings in Washington DC from January 5th to January 8th, 2009, and giving talks on 2-groups and groupoidification. Who else will be in town?
**Urs Schreiber:**Groupoidification from sigma-Models? (December 30, 2008)- On constructing sigma-models from differential nonabelian cocycles and how this gives rise to methods of groupoidification.
**John Baez:**Groupoidification Made Easy (December 26, 2008)- Now you can learn about groupoidification in a paper that has proofs of all the most basic results!
**John Baez:**Tilings (December 24, 2008)- Here are some pretty pictures of tilings, suitable for use as computer wallpaper.
**John Baez:**Infinite-Dimensional Representations of 2-Groups (December 24, 2008)- Here’s a new paper on infinite-dimensional representations of 2-groups — representations on ‘measurable categories’.
**John Baez:**Science and the Environment (December 24, 2008)- Scientists may soon be running science policy in the US. What should we do?
**David Corfield:**Bridge Building (December 23, 2008)- Might the cohomology of dynamical systems provide a meeting ground for researchers on the ‘combinatorics’ side of mathematics, and those on the ‘theory-building’ side?
**John Baez:**Higher Structures in Göttingen - Part II (December 22, 2008)- On February 5-7 2009 there will be a workshop in Göttingen on Higher Structures in Topology and Geometry.
**David Corfield:**The Microcosm Principle (December 19, 2008)- The microcosm principle finds aspects of the small reflected in the large.
**John Baez:**Super Version of 2-Plectic Geometry for Classical Superstrings? (December 16, 2008)- Here are some basic questions about describing the classical superstring using the ‘super’ analogue of multisymplectic geometry, if such a thing exists.
**David Corfield:**What to Make of Mathematical Difficulties (December 16, 2008)- How to make the most of obstructions to a theory
**John Baez:**This Week’s Finds in Mathematical Physics (Week 273) (December 15, 2008)- Read more about the geysers of Enceladus. Hear the history of mineral evolution, from chondrites to the Big Splat, the Late Heavy Bombardment, the Great Oxidation Event, and Snowball Earth… to now. Then, learn about Pontryagin duality.
**David Corfield:**The Status of Coalgebra (December 9, 2008)- Is coalgebra just as important as algebra?
**John Baez:**Science Citation Index (December 9, 2008)- The Science Citation Index is increasingly being used to evaluate research. There are lots of reasons why this is a bad idea.
**John Baez:**A Quick Algebra Quiz (December 9, 2008)- Here’s a quick test of your mathematical reflexes.
**John Baez:**Smooth Structures in Ottawa (December 7, 2008)- There will be a conference on Smooth Structures in Logic, Category Theory and Physics at the University of Ottawa on May 1-3, 2009.
**Urs Schreiber:**Question on Infinity-Yoneda (December 4, 2008)- What is known about the Yoneda lemma in a higher categorical context?
**Urs Schreiber:**Zhu on Lie’s Second Theorem for Lie Groupoids (December 3, 2008)- On Lie integration of Lie algebroids.
**David Corfield:**Tom Leinster in The Reasoner (November 30, 2008)- Tom Leinster interviewed for The Reasoner
**John Baez:**This Week’s Finds in Mathematical Physics (Week 272) (November 29, 2008)- See what the Cassini probe saw as it shot through the Enceladus plumes, see what happens when the Io flux tube hits Jupiter, learn two new properties of the number 6, and discover the wonders of standard Borel spaces, commutative von Neumann algebras, and Polish groups.
**John Baez:**Groupoidfest 08 - A Brief Report (November 28, 2008)- A brief report on the 2008 Groupoidfest held at UCR, with an emphasis on
*groupoidification*. **Urs Schreiber:***n*Lab (November 28, 2008)- We have set up a Wiki accompanying the n-Category Cafe.
**David Corfield:**Coalgebraically Thinking (November 27, 2008)- How coalgebra differs from algebra.
**Urs Schreiber:**Mathematics, Physics and Philosophy (November 26, 2008)- An essay, or the like, on motivations for interdisciplinary research at the boundary of math, physics and philosophy.
**Urs Schreiber:**Bär on Fiber Integration in Differential Cohomology (November 26, 2008)- On fiber integration in differential cohomology and the notion of generalized smooth spaces used for that.
**David Corfield:**Beyond the Blog (November 25, 2008)- How to make best use of blog material.
**John Baez:**Categorification in Glasgow (November 24, 2008)- There will be a workshop on categorification in Glasgow from April 13th to April 18th, 2009.
**David Corfield:**Mathematical Robustness (November 20, 2008)- Does the ‘robust’ appearance of mathematical entities indicate reality?
**Urs Schreiber:***Higher and Graded Geometric Structures*in Göttingen (November 19, 2008)- A workshop on higher bundles, gerbes, groupoids and supermanifolds.
**David Corfield:**Fourth Categories, Logic and Physics Workshop (November 18, 2008)- Categories, Logic and Physics workshop announcement.
**Urs Schreiber:**Local Nets and Co-Sheaves (November 14, 2008)- Co-sheaf condition (codescent) for Haag-Kastler nets of local quantum observables?
**Urs Schreiber:**Frobenius Algebras and the BV Formalism (November 14, 2008)- Bruce Bartlett is looking at the latest article by Cattaneo and Mnev on BV-quantization of Chern-Simons theory.
**Urs Schreiber:**Baković on*Bigroupoid 2-Torsors*(November 13, 2008)- Bakovic on G-principal bundles for G a bigroupoid.
**Urs Schreiber:**Talk:*Local Nets from Parallel Transport 2-Functors*(November 12, 2008)- A talk on the relation between AQFT and extended functorial QFT.
**John Baez:**Locally Compact Hausdorff Abelian Groups (November 12, 2008)- Learn the Principal Structure Theorem for locally compact Hausdorff abelian groups — the kind of groups that show up in Pontryagin duality.
**John Baez:**Variations on Pontryagin Duality (November 8, 2008)- Here are some basic questions about the process of taking a locally compact abelian group
*A*to the group of homomorphisms from*A*to the invertible complex numbers. **David Corfield:**Entropy, Diversity and Cardinality (Part 2) (November 7, 2008)- Continuation of a discussion of entropy and cardinality and its relevance to ecology
**Urs Schreiber:**Lie 3-Algebras on the Membrane (?) (November 6, 2008)- Recently a trinary bracket appears in the study of the supermembrane which is sometimes addressed as a homotopy algebraic structure.
**David Corfield:**The Internal Language of a 2-Topos (November 6, 2008)- Extracting the internal language from a 2-topos
**David Corfield:**Maximum Entropy Distributions (November 2, 2008)- Why are familiar distributions so often maximum shannon entropy distributions for their class?
**Urs Schreiber:***Higher Structures in Math and Physics*in Lausanne (November 2, 2008)- A workshop in Lausanne on Higher Structures in Mathematics and Physics
**Urs Schreiber:**Twisted Differential Nonabelian Cohomology (October 30, 2008)- Work on theory and applications of twisted nonabelian differential cohomology.
**John Baez:**Google Books — More Open Access? (October 28, 2008)- A settlement has been reached which may give us more free access to Google Books.
**John Baez:**Lie Theory Through Examples 4 (October 28, 2008)- Classifying representations using weights.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 271) (October 26, 2008)- Read about massive volcanic eruptions on Jupiter’s moon Io, allotropes of sulfur, quasicrystals in various dimensions, Jeffrey Morton’s extension of the “groupoidification” program, and Stephen Summers’ review of new work on constructive quantum field theory!
**John Baez:**Open Access at the University of California (October 24, 2008)- News about open access at the University of California.
**John Baez:**Groupoidfest 08 (October 24, 2008)- Some talks at the Groupoidfest being held in Riverside on November 22nd-23rd, 2008.
**Urs Schreiber:**Hopf Algebraic Renormalization (October 23, 2008)- The starting point and basic idea of combinatorial Hopf algebraic techniques in perturbative quantum field theory.
**David Corfield:**Reviewing Ruelle’s Book (October 23, 2008)- Announcing my review of Ruelle’s ‘The Mathematician’s Brain’.
**John Baez:**What is Categorification? (October 22, 2008)- The meaning of the word ‘categorification’, and some key examples.
**Urs Schreiber:**Codescent and the van Kampen Theorem (October 21, 2008)- On codescent, infinity-co-stacks, fundamental infinity-groupoids, natural differential geometry and the van Kampen theorem
**David Corfield:**John McKay Visits Kent (October 21, 2008)- What I took from John McKay’s talk on finite simple groups.
**John Baez:**Lie Theory Through Examples 3 (October 21, 2008)- Finally the concept of ‘weight lattice’ rears its ugly head.
**Urs Schreiber:**Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack (October 20, 2008)- Hepworth on the idea of volume forms on a stack in terms of sections of 2-vector bundles.
**John Baez:**Categorification in New Scientist (October 20, 2008)- Richard Elwes at
*New Scientist*has written an article on knot theory that mentions categorification. **Urs Schreiber:**Talk in Göttingen:*Second Nonabelian Differential Cohomology*(October 19, 2008)- A talk on second nonabelian differential cohomology.
**John Baez:**Morton on 2-Vector Spaces and Groupoids (October 19, 2008)- Jeffrey Morton has a paper that turns groupoids into 2-vector spaces — a key step toward constructing the Dijkgraaf–Witten model as an extended TQFT.
**David Corfield:**Entropy, Diversity and Cardinality (Part 1) (October 16, 2008)- Entropy in relation to biodiversity and categories
**Urs Schreiber:**String- and Fivebrane-Structures (October 14, 2008)- A new article on String- and Fivebrane structures and some previous articles on Fivebrane structures.
**Urs Schreiber:**Freely Generated ω-Categories (October 14, 2008)- The notion of strict infinity-categories which are “freely generated”.
**John Baez:**The Nature of Time (October 13, 2008)- The Foundational Questions Institute is having an essay contest on the nature of time.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 270) (October 12, 2008)- See lava on Jupiter’s moon Io. Hear about Greg Egan’s new novel. And then, learn about some little-known interactions between the numbers 5, 8, 12, and 24.
**Urs Schreiber:**Semistrict Infinity-Categories and ω-Semi-Categories (October 8, 2008)- On strict infinity categories with weak identities.
**David Corfield:**New Directions in the Philosophy of Mathematics (October 8, 2008)- To celebrate the founding of MIMS, the mathematics department of the recently unified Manchester University, it was proposed that various workshops named ‘New Directions in…’ be run. They kindly agreed to allow Alexandre Borovik and me to organise one…
**Urs Schreiber:**Yet Another Model ω-Question (October 7, 2008)- On the relation between the model category structure on smooth infinity-groupoids and on differential algebras.
**John Baez:**Lie Theory Through Examples 2 (October 7, 2008)- Moving on up a dimension, now let’s look at the A
_{3}lattice. This arises naturally from the group SU(4), but you’ve also seen it in grocery stores if you ever paid attention to stacks of oranges. **David Corfield:**The Blog of Fun (October 5, 2008)- The ‘field with one element’ has been honoured by a great accolade. As announced here, it has been awarded a blog all to itself. Not bad for an entity with dubious existence credentials….
**Urs Schreiber:**More Model ω-Questions (October 2, 2008)- More questions on model category theory in general and the model structure on omega-categories in particular.
**David Corfield:**Mathematical Reality (October 1, 2008)- Some thoughts on mathematical reality as pieces of mathematics inevitably encountered in different ways.
**Urs Schreiber:**(ω+ω)-Categories (?) (September 30, 2008)- A tentative suggestion for weak oo-categories.
**John Baez:**Quantum Theory and Analysis (September 30, 2008)- Some lecture notes for a class on Hilbert spaces, self-adjoint operators, and their applications to quantum theory.
**John Baez:**Lie Theory Through Examples 1 (September 30, 2008)- The beginning of a course on Lie theory through examples.
**John Baez:**OctoberFest 08 (September 27, 2008)- This October, the annual Canadian conference on categories will be at Concordia University in Montreal.
**Urs Schreiber:**Some Model ω-Questions (September 27, 2008)- The model structure on omega-categories should induce a model structure on omega-category valued presheaves. What can be said about this?
**David Corfield:**Mathematical Miniatures (September 26, 2008)- Short essays by Beno Eckmann
**Urs Schreiber:**Group Cocycles and Simplices (September 22, 2008)- Question on simplicial group cocycles.
**Urs Schreiber:**Back and Catching Up (September 21, 2008)- A list of things to read.
**David Corfield:**The Volume of a Differentiable Stack (September 15, 2008)- Weinstein’s generalisation of the Baez-Dolan groupoid cardinality concept to stacks.
**John Baez:**5 (September 12, 2008)- A talk on special properties of the number 5.
**John Baez:**24 (September 12, 2008)- A talk on my favorite number of all: 24.
**Urs Schreiber:**Planar Algebras, TFTs with Defects (September 11, 2008)- On planar algebras interpreted as 2D TFTs with defect lines.
**John Baez:**New Structures for Physics II (September 10, 2008)- You can help review some more papers for Bob Coecke’s
*New Structures for Physics*. **David Corfield:**Who are Various? (September 10, 2008)- Workshop announcement and an enquiry about a reference made by Saunders Mac Lane
**David Corfield:**Reliability (September 9, 2008)- A response to an article on the unreliability of the mathematical literature.
**Urs Schreiber:**Smooth Differential Graded Algebra? (September 9, 2008)- What is the right definition of generalized smooth differential graded-commutative algebras?
**Urs Schreiber:**Equivariant deRham Cohomology and Action Lie Algebroids (September 7, 2008)- The relation between equivariant deRham cohomology and the Weil algebra of the corresponding action Lie algebroid.
**John Baez:**8 (September 7, 2008)- A talk on one of my favorite numbers: 8.
**John Baez:**Military Research (September 6, 2008)- What sort of research is the US military funding today?
**Urs Schreiber:**Waldorf on*Transport Functors and Connections on Gerbes*(September 6, 2008)- A talk on parallel 2-transport.
**John Baez:**Comparative Smootheology, III (September 3, 2008)- The third episode in our continuing comparison of various frameworks for differential geometry.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 269) (August 31, 2008)- See more of Jupiter’s moon Io — and learn about honeycombs, the work of Kelvin, the Weaire-Phelan structure, and gas clathrates.
**David Corfield:**Imperfections, Ambiguities and Physics (August 30, 2008)- Can Lautman’s concept of the dialectical relationship between perfection and imperfection be realised in physics?
**John Baez:**New Structures for Physics I (August 29, 2008)- Please comment on two chapters of a forthcoming book edited by Bob Coecke: ‘Introduction to categories and categorical logic’ by Abramsky and Tzevelekos, and ‘Categories for the practicing physicist’ by Coecke and Paquette.
**John Baez:**Henri Cartan, July 8, 1904 - August 13, 2008 (August 22, 2008)- Henri Cartan, a founding member of Bourbaki, died August 13th at the age of 104.
**David Corfield:**Melliès on Functorial Boxes (August 22, 2008)- Paul-André Melliès has a tutorial on incorporating functors
*between*categories into string diagrams… and applications to logic. **John Baez:**The Weaire–Phelan Structure (August 21, 2008)- The best known solution to an old math problem posed by Kelvin now graces the National Aquatic Center at the Olympic games in Beijing.
**Urs Schreiber:**Homotopical Algebra with Applications to Mathematical Physics (August 21, 2008)- A workshop on homotopical methods in mathematical physics.
**John Baez:**Polish Spaces (August 21, 2008)- Are second-countable locally compact Hausdorff spaces always homeomorphic to complete separable metric spaces?
**David Corfield:**Freed on Chern-Simons (August 21, 2008)- Daniel Freed has come out with an expository paper on Chern-Simons theory.
**John Baez:**Elvis Zap does the Calculus Rap (August 20, 2008)- Professor Elvis Zap explains the rules of differential calculus.
**John Baez:**Spivak on Derived Manifolds (August 19, 2008)- David Spivak has an interesting thesis on ‘derived differential geometry’.
**Urs Schreiber:**Bergman on Infinity-Vector Bundles Coupled to Topological Strings (August 17, 2008)- topological string, B-model, derived category, B-brane, infinity-vector space, infinity connection
**Urs Schreiber:**Differential Graded Clifford Algebra (August 17, 2008)- On Clifford-defomrations of Chevalley-Eilenberg and Weil algebras.
**Urs Schreiber:**Connections on Nonabelian Gerbes and their Holonomy (August 15, 2008)- An article on transport 2-functors.
**David Corfield:**Lautman on Reciprocity (August 14, 2008)- Albert Lautman has a paper on the use of analysis in number theory — with a special emphasis on “reciprocity”.
**John Baez:**Bimodules Versus Spans (August 14, 2008)- Some puzzles about bimodules in the opposite of the category of sets.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 268) (August 6, 2008)- See a marvelous view of Io, and then learn more about Frobenius algebras than you probably wanted to know — and a bit about modular tensor categories and the mathematics of music, too!
**John Baez:**Visiting Grothendieck (August 6, 2008)- The story of someone who tried to visit Alexander Grothendieck.
**John Baez:**Categorified Symplectic Geometry and the Classical String (August 2, 2008)- In this new version of our paper, we systematically explain how n-dimensional field theories give n-plectic manifolds. We also say how a B field affects the 2-plectic structure for a string.
**John Baez:**Getting Started Early (July 31, 2008)- The man who had Ph.D. students before he was born.
**Urs Schreiber:**Pre- and Postdictions of the NCG Standard Model (July 30, 2008)- Some predictions and postdictions derived from the NCG Kaluza-Klein standard model.
**David Corfield:**Category Theory and Model Theory (July 29, 2008)- Category theory and model theory
**John Baez:**Light Mills (July 28, 2008)- The Crookes radiometer is also known as a ‘light mill’ — a little glass bulb with a windmill in it, with vanes black on one side and white on the other. It puzzled Reynolds, Maxwell and even Einstein. Do we really understand it yet?
**John Baez:**Causality in Discrete Models of Spacetime (July 27, 2008)- Gavin Wraith has some questions about causality in discrete spacetimes described as simplicial complexes.
**John Baez:**Categories, Logic and Foundations of Physics in Oxford (July 25, 2008)- On August 23rd and 24th, there will be a conference on Categories, Logic and Foundations of Physics at Oxford.
**David Corfield:**Real versus Complex Numbers (July 24, 2008)- Comparing the reals and the complex numbers
**John Baez:**This Week’s Finds in Mathematical Physics (Week 267) (July 23, 2008)- In “week267” see the tilings of the Alhambra and learn about wallpaper groups, 17 wallpaper groups, their corresponding 2d orbifolds, the role of 2-groups as symmetries of orbifolds, the work of Carrasco and Cegarra on hypercrossed complexes, and the work of João Faria Martins on the fundamental 2-group of a 2-knot.
**David Corfield:**Girard on the Limitations of Categories (July 23, 2008)- Girard on categories
**David Corfield:**Hierarchy and Emergence (July 18, 2008)- Emergence in hierarchies
**Urs Schreiber:**Some ω-Questions (July 18, 2008)- Questions on omega-categorical constructions in the context of descent and cohomology.
**Urs Schreiber:**News on Measures on Groupoids? (July 17, 2008)- Benjamin Bahr apparently thought about measures on groupoids of connections.
**John Baez:**Theorems Into Coffee IV (July 12, 2008)- The first Theorems into Coffee prize is awarded. Read about Steve Lack’s work on PROPs, and try your hand at the latest Theorems into Coffee challenge.
**Urs Schreiber:**Talk on*AQFT from FQFT and Applications*(July 10, 2008)- On local nets constructed from transport 2-functors and examples relating to lattice models, Hopf spin chains, asymptotic inclusion of subfactors. And some remarks on the relation between conformal nets and vertex operator algebras.
**Urs Schreiber:**Basics of Poisson Reduction and BV, I (July 7, 2008)- Basics of symplectic reduction and Poisson reduction.
**David Corfield:**Sphere Eversion (July 7, 2008)- Everting the sphere
**David Corfield:**A Small Observation (July 3, 2008)- A duality in 2-vector spaces
**Urs Schreiber:**The Manifold Geometries of QFT, II (Suijlekom on Renormalization, Hopf Algebra and BV-Formalism) (July 1, 2008)- Walter Suijlekom explains a relation between Connes-Kreimer Hopf algebras of Feynman diagrams with the master equation in BV-formalism.
**Urs Schreiber:**Block on L-∞ Module Categories (June 30, 2008)- On Jonathan Block’s concept of modules over differential graded algebras.
**Urs Schreiber:**The Manifold Geometries of QFT, I (June 30, 2008)- Some notes on the first day at “The manifold geometries of QFT” at the MPI in Bonn.
**David Corfield:**Lerman on Orbifolds (June 30, 2008)- Lerman on orbifolds
**David Corfield:**Michael Polanyi and Personal Knowledge (June 28, 2008)- Polanyi on personal knowledge
**Urs Schreiber:**AQFT from Lattice Models (?) (June 27, 2008)- On lattice AQFT models such as Hopf spin chains and their derivation of 2-C*-algebra valued 2-functors.
**Urs Schreiber:**Eli Hawkins on Geometric Quantization, II (June 27, 2008)- Eli Hawkins explains his method of getting a quantum algebra from the convolution algebra of sections on a symplectic groupoid.
**Urs Schreiber:**Category Algebras (June 26, 2008)- On an interpretation of the category algebra of a category in terms of pull-push of finite sets through correspondence spaces.
**David Corfield:**Tim Porter on Formal Homotopy Quantum Field Theories and 2-Groups (June 24, 2008)- Tim Porter on HQFT
**Urs Schreiber:**Behrang Noohi on Butterflies and Morphisms Between Weak 2-Groups (June 23, 2008)- A talk by Behrang Noohi.
**Urs Schreiber:**Aldrovandi on Non-Abelian Gerbes and 2-Bundles (June 23, 2008)- A talk by Aldrovandi on different perspectives on gerbes and 2-bundles.
**Urs Schreiber:**Landsman on Quantization of Poisson Algebras Associated to Lie Algebroids (June 22, 2008)- A quick review of Landsman’s result on strict deformation quantization of Poisson manifolds dual to Lie algebroids: the quantum algebra is nothing but the groupoid algebra of the Lie groupoid integrating the Lie algebroid.
**Urs Schreiber:**Eli Hawkins on Geometric Quantization, I (June 20, 2008)- Some basics and some aspects of geometric quantization. With an emphasis on the geometric quantization of duals of Lie algebras and duals of Lie algebroids.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 266) (June 20, 2008)- Read about Jupiter’s moon Io, the Pythagorean tuning system, and the categorical groups conference now taking place in Barcelona.
**John Baez:**Fundamental 2-Groups and 2-Covering Spaces (June 19, 2008)- You can see the slides of David Roberts’ talk on the fundamental 2-group of a topological space, and the corresponding notion of ‘2-covering space’.
**Urs Schreiber:**Schommer-Pries on Classification of 2-Dimensional Extended TFT (June 18, 2008)- Chris Schommer-Pries classifies 2-functors from a 2-category of 2-dimensional cobordisms.
**Urs Schreiber:**Teleman on Topological Construction of Chern-Simons Theory (June 17, 2008)- A talk by Constant Teleman on extended Chern-Simons QFT and what to assign to the point.
**David Corfield:**The Mathematical Sublime (June 17, 2008)- The sublime
**John Baez:**Computation and the Periodic Table (June 14, 2008)- The analogies between physics, topology, logic and computer science, visible so clearly with the help of symmetric monoidal closed categories, are just the tip of a larger iceberg involving $n$-categories. The Periodic Table seems to be a useful guide here.
**Urs Schreiber:**An Exercise in Groupoidification: The Path Integral (June 13, 2008)- A remark on the path integral in view of groupoidification and Sigma-model quantization.
**Urs Schreiber:**A Groupoid Approach to Quantization (June 12, 2008)- On Eli Hawkins’ groupoid version of geometric quantization.
**David Corfield:**Klein 2-Geometry XI (June 12, 2008)- categorified geometry
**John Baez:**Categories, Logic and Foundations of Physics: The Videos (June 9, 2008)- Now videos are available for the 2nd conference on Categories, Logic and Foundations of Physics.
**John Baez:**2-Groups in Barcelona (June 8, 2008)- There’s a workshop on categorical groups in Barcelona during June 16-20, 2008.
**John Baez:**Farey Sequences and the Stern-Brocot Tree (June 7, 2008)- An approach to modular curves based on Farey sequences and the Stern-Brocot tree.
**Urs Schreiber:**Urban Myths in Contemporary Cosmology (June 6, 2008)- On urban-myth-like memes in the contemporary theoretical cosmology community.
**David Corfield:**Dumbing Down (June 4, 2008)- Mathematics exams for 16 year olds are getting easier, it is claimed. It’s fairly easy to check for yourself. Take a look at the Arithmetic, Algebra and Geometry papers from 1959 and compare with a contemporary specimen GCSE paper. Even…
**John Baez:**Classical String Theory and Categorified Symplectic Geometry (June 2, 2008)- Just as any symplectic manifold gives a Lie algebra of observables, any 2-plectic manifold gives a Lie 2-algebra of observables. This shows up in string theory!
**Urs Schreiber:**HIM Trimester on Geometry and Physics, Week 4 (May 29, 2008)- Talk in Stanford on nonabelian differential cohomology.
**David Corfield:**Manin on Foundations (May 28, 2008)- Manin on mathematical foundations
**John Baez:***Cahiers*Free Online! (May 27, 2008)- The Ehresmann’s famous journal is now free online.
**John Baez:**Double Categories — Warm and Cuddly? (May 27, 2008)- Abstract of a talk on double categories by Ross Street.
**John Baez:**Number Theory on YouTube? (May 26, 2008)- Number theory abstracts on YouTube?
**John Baez:**This Week’s Finds in Mathematical Physics (Week 265) (May 26, 2008)- Read about Europa, the Pythagorean pentagram, Bill Schmitt’s work on Hopf algebras in combinatorics, the magnum opus of Aguiar and Mahajan, and quaternionic analysis.
**Urs Schreiber:**Workshop: Non-Commutative Constructions in Arithmetic and Geometry (May 22, 2008)- Conference in London on non-commutative constructions in arithmetic and geometry
**Urs Schreiber:**Hopkins-Lurie on Baez-Dolan (May 20, 2008)- Hopkins and Lurie give a proof of the Baez-Dolan hypothesis on tangles and extended TQFT.
**Urs Schreiber:**Relation between AQFT and Extended Functorial QFT (May 20, 2008)- On how to obtain from any extended d-dimensional QFT a local net of observables.
**David Corfield:**Ambiguity Theory (May 19, 2008)- Galois theory as the study of ambiguity
**Urs Schreiber:**Integrals and Valuations using Geometric Logic (May 19, 2008)- Bas Spitters on the theory of integration in a topos.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 264) (May 19, 2008)- Learn about Phobos, homotopy groups of spheres, and theta functions.
**Urs Schreiber:**Harrison’s Geometric Calculus (May 18, 2008)- On Harrison’s generalized formalism for differential calculus and differential geometry.
**Urs Schreiber:**Electric-Magnetic-Duality and Hodge Duality Extended to Differental Cocycles (May 17, 2008)- On the electric-magnetic dual formulation of higher abelian Yang-Mills theory.
**John Baez:**Convenient Categories of Smooth Spaces (May 17, 2008)- Chen spaces and Souriau’s diffeological spaces are two great contexts for differential geometry. Alex Hoffnung and his thesis advisor just wrote a paper studying these in detail.
**John Baez:**Theorems Into Coffee III (May 15, 2008)- What is the symmetric monoidal category of finite sets and partially defined functions the PROP for? Prove your answer and win some coffee!
**Urs Schreiber:**HIM Trimester Geometry and Physics, Week 1 (May 14, 2008)- On vertex operator algebras, operads and Segals QFT axioms. On integration over supermanifolds, supergroupoids and the two notions of supercategories.
**Urs Schreiber:**E_{8}Quillen Superconnection (May 10, 2008)- On Quillen superconnections on E8-bundles and on the mathematical interpretation of the connection appearing in an article by A. Lisi.
**Urs Schreiber:**On Lie oo-Theory (May 8, 2008)- Talk on the Lie infinity-algebroid perspective on the BRST complex and its relation to the integrated picture of action infinity-groups.
**David Corfield:**Pernicious Symbolization (May 8, 2008)- Ryle against Carnap
**Urs Schreiber:**Integrability of Lie Brackets (May 7, 2008)- Crainic and Fernandes on integration of Lie algebroids to Lie groupoids.
**Urs Schreiber:**Ambimorphic? (May 6, 2008)- On adjunctions between spaces and n-groupoids induced by homming out of the fundamental n-groupoid.
**David Corfield:**Quaternionic Analysis (May 6, 2008)- Quaternionic analysis
**John Baez:**Theorems Into Coffee II (May 6, 2008)- Another coffee challenge: prove that matrices of integers are morphisms in the PROP for commutative and cocommutative Hopf algebras!
**John Baez:**Theorems Into Coffee (May 4, 2008)- Prove this theorem, and you’ll get some coffee!
**Urs Schreiber:**FQXi (May 4, 2008)- The Foundational Questions Institute runs its second grant competition.
**David Corfield:**Questions on 2-covers (April 30, 2008)- 2-covers
**David Corfield:**Returning to Lautman (April 29, 2008)- I mentioned in an earlier post that Albert Lautman had a considerable influence on my decision to turn to philosophy. I recently found out that his writings have been gathered together and republished as Les mathématiques, les idées et le…
**Urs Schreiber:**Dual Formulation of String Theory and Fivebrane Structures (April 28, 2008)- An article which discusses lifts through the 7-fold connected cover of the structure group of the tangent bundle in the context of electric-magnetic duality in string theory.
**Urs Schreiber:**Charges and Twisted Bundles, IV: Anomaly Cancellation (April 27, 2008)- How the fermionic anomaly may cancel against the charge anomaly in higher gauge theory: the Green-Schwarz mechanism
**Urs Schreiber:**Charges and Twisted Bundles, III: Anomalies (April 25, 2008)- On the anomalies arising in (higher) gauge theories in the presence of electric and magnetic charges.
**Urs Schreiber:**Thoughts (Mostly on Super ∞-Things) (April 22, 2008)- Thoughts while travelling and talking.
**Urs Schreiber:**Higher Hochschild Cohomology and Differential Forms on Mapping Spaces (April 18, 2008)- On “brane topology”, the higher order generalization of Hochschild cohomology. And on Chen iterated integrals.
**John Baez:**Comparative Smootheology, II (April 17, 2008)- Comparing Chen’s ‘differentiable spaces’ and Souriau’s ‘diffeological spaces’.
**Urs Schreiber:**A Topos for Algebraic Quantum Theory (revised) (April 15, 2008)- Bas Spitters on topos-theory of noncommutative C*-algebras and quantum physics.
**David Corfield:**Klein 2-Geometry X (April 15, 2008)- Acting on groups
**Urs Schreiber:**Sigma-Models and Nonabelian Differential Cohomology (April 11, 2008)- Notes on nonabelian differential cohomology and its application to classical and qantum parallel transport.
**David Corfield:**Categorical Sheaves (April 9, 2008)- Toën and vezzosi on categorical sheaves
**David Corfield:**Petitition to Save USQ Mathematics (April 9, 2008)- Terry Tao on saving USQ Maths
**John Baez:**This Week’s Finds in Mathematical Physics (Week 263) (April 6, 2008)- John Thompson and Jacques Tits won the 2008 Abel Prize for their work on group theory. Find out what they did.
**David Corfield:**2-Structure Types (April 2, 2008)- 2-structure types
**John Baez:**A Puzzle About Blueshifts (April 1, 2008)- A puzzle about blueshifts for rotating black holes, posed by Greg Egan.
**Urs Schreiber:**Limits and Push-Forward (March 31, 2008)- Question on the relation between push-forward of functors and (indexed) limits.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 262) (March 30, 2008)- Learn about the Southern Ring Nebula, the frosty dunes of Mars, quantum technology in Singapore, atom chips, graphene transistors, nitrogen-vacancy pairs in diamonds, a
new construction of e
_{8}, and a categorification of sl(2). **John Baez:**Test Your Singlish (March 29, 2008)- Guess the meaning of some words in Singlish.
**David Corfield:**Categorified Quantum Groups (March 27, 2008)- categorifying quantum groups
**Urs Schreiber:**What Has Happened So Far (March 27, 2008)- A review of one of the main topics discussed at the Cafe: Sigma-models as the pull-push quantization of nonabelian differential cocycles.
**Urs Schreiber:**Nonabelian Differential Cohomology in Street’s Descent Theory (March 22, 2008)- A discussion of differential nonabelian cocycles classifying higher bundles with connection in the context of the general theory of descent and cohomology with coefficients in infnity-category valued presheaves as formalized by Ross Street.
**John Baez:**Groupoidfest in Riverside (March 21, 2008)- The next Groupoidfest is at UCR, November 22-23 2008.
**Urs Schreiber:**Crossed Menagerie (March 21, 2008)- Detailed notes by Tim Porter on “crossed gadgetry and cohomology in algebra and topology”.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 261) (March 20, 2008)- Learn about the Engraved Hourglass Nebula… and read an ode to the number 3. Also: try your hand at some trefoil puzzles.
**David Corfield:**The World of L (March 17, 2008)- Breaking news in number theory
**Urs Schreiber:**Slides:*On Nonabelian Differential Cohomology*(March 13, 2008)- On the notion of nonabelian differential cohomology.
**Urs Schreiber:**Chern-Simons Actions for (Super)-Gravities (March 12, 2008)- On Chern-Simons actions for (super-)gravity.
**John Baez:**Following Singapore’s Lead (March 12, 2008)- A Los Angeles elementary school drastically improves its math teaching… by following the lead of Singapore.
**David Corfield:**A Strange Link (March 11, 2008)- Tim Porter on global actions
**John Baez:**Physics, Topology, Logic and Computation: a Rosetta Stone (March 11, 2008)- Toward a general theory of systems and processes.
**John Baez:**Learning to Love Topos Theory (March 9, 2008)- Steve Vickers on topos theory.
**John Baez:**Some Puzzles (March 8, 2008)- Three puzzles to keep you entertained.
**Urs Schreiber:**Space and Quantity (March 6, 2008)- Notes on spaces and smooth spaces, function algebras and smooth function algebras.
**David Corfield:**Worrying About 2-Logic (March 6, 2008)- More on 2-logic — some possible problems.
**Urs Schreiber:**Sections of Bundles and Question on Inner Homs in Comma Categories (March 4, 2008)- On inner homs in comma categories, motivated from a description of spaces of sections of bundles in terms of such.
**Urs Schreiber:**Charges and Twisted n-Bundles, II (March 4, 2008)- Rephrasing Freed’s action functional for differential cohomology in terms of L-oo connections in a simple toy example.
**Urs Schreiber:**Infinity-Groups with Specified Composition (March 3, 2008)- On infinity-groups and infinity-categories with specified composition, and their closedness.
**David Corfield:**A Deep Sense of Miserable Ignorance (March 3, 2008)- Peirce on pedagogy
**John Baez:**Kim on Fundamental Groups in Number Theory (March 1, 2008)- Minhyong Kim on Diophantine geometry and the fundamental group.
**John Baez:**Computer Scientists Needed Now (March 1, 2008)- John Baez begs for help on the computation section of his ‘Rosetta Stone’ paper connecting physics, topology, logic and computation.
**Urs Schreiber:**Charges and Twisted n-Bundles, I (February 29, 2008)- Generalized charges are very well understood using generalized differential cohomology. Here I relate that to the nonabelian differential cohomology of n-bundles with connection.
**David Corfield:**Peirce on Mathematics (February 28, 2008)- Charles Peirce on mathematics.
**Urs Schreiber:**(Generalized) Differential Cohomology and Lie Infinity-Connections (February 27, 2008)- On generalized differential cohomology and its relation to infinity-parallel transport and Lie-infinity connections.
**John Baez:**New Hire at UCR (February 27, 2008)- Julia Bergner has accepted a tenure-track position at UCR. She works on the homotopy theory of homotopy theories.
**Urs Schreiber:**Impressions on Infinity-Lie Theory (February 26, 2008)- Thoughts on infinity Lie theory.
**Urs Schreiber:**What I learned from Urs (February 26, 2008)- Bruce Bartlett talks about some aspects of the program of systematically understanding the quantization of Sigma-models in terms of sending parallel transport n-functors to the cobordism representations which encode the quantum field theory of the n-particles charged under them.
**David Corfield:**A Question or Two (February 25, 2008)- Is there a categorified spectrum?
**Urs Schreiber:**Lurie on Extended TQFT (February 21, 2008)- Jacob Lurie on extended TQFT and the Baez-Dolan Cobordism Hypothesis
**John Baez:**Logicians Needed Now (February 21, 2008)- Mike Stay and I are writing a paper… and we need help from logicians!
**John Baez:**Kostant on E_{8}(February 20, 2008)- Videos and lecture notes of Bertram Kostant’s talk on E
_{8}and the algebra of the Standard Model. **Urs Schreiber:**Categories, Logic and Physics in London, II (February 19, 2008)- Second workshop on “Categories, Logic and Foundations of Physics” in London.
**John Baez:**Harvard Research Free Online (February 18, 2008)- Harvard arts and sciences faculty vote to make their research free online!
**Urs Schreiber:**Slides:*L-Infinity Connections and Applications*(February 15, 2008)- Talks on L-infinity algebra connections.
**David Corfield:**2-Galois and 2-Logic (February 13, 2008)- Categorifying logic
**John Baez:**Verity on ∞-Categories From Topology (February 12, 2008)- Dominic Verity talks about a weak ∞-category of cobordisms.
**David Corfield:**Lautman Conference (February 12, 2008)- Lautman conference
**John Baez:**Geometric Representation Theory (Lecture 25) (February 12, 2008)- Groupoidifying the commutation relations between annihilation and creation operators in quantum mechanics. An in-class experiment demonstrating these relations.
**Urs Schreiber:**Construction of Cocycles for Chern-Simons 3-Bundles (February 12, 2008)- On how to interpret the geometric construction by Brylinksi and McLaughlin of Cech cocycles classified by Pontrjagin classes as obstructions to lifts of G-bundles to String(G)-2-bundles.
**Urs Schreiber:**Question on Smooth Functions (February 11, 2008)- On morphisms of algebras of smooth functions coming from pullback along smooth maps.
**John Baez:**Metric Spaces (February 9, 2008)- Tom Leinster on the cardinality of a metric space, thought of as an enriched category — and an interesting relation to geometric measure theory.
**Urs Schreiber:**Smooth 2-Functors and Differential Forms (February 6, 2008)- An article on the relation between smooth 2-functors with values in strict 2-groups, and an outline of the big picture that this sits in.
**Urs Schreiber:**Chern-Simons States from L-infinity Bundles, III: States over the Circle (February 4, 2008)- On computing the states of Chern-Simons theory over the circle from the L-infinity algebraic model of the Chern-Simons 3-bundle over BG.
**David Corfield:**Albert Lautman (February 4, 2008)- The philosophy of Albert Lautman
**David Corfield:**Virtually Real or Really Virtual (February 2, 2008)- If NASA can have a presence in Second Life with their CoLab project, maybe we at the Café should be thinking about the next step….
**John Baez:**Modular Forms (February 1, 2008)- What are modular forms like beyond level 1? How about level 2, for starters?
**Urs Schreiber:**States of Chern-Simons Theory (February 1, 2008)- A list of selected literature discussing Chern-Simons theory and its space of states.
**John Baez:**Geometric Representation Theory (Lecture 24) (January 31, 2008)- How to groupoidify the harmonic oscillator.
**Urs Schreiber:**L-infinity Associated Bundles, Sections and Covariant Derivatives (January 30, 2008)- Associated L-infinity structures are obtained from Lie action infinity-algebroids, leading to a concept of sections and covariant derivatives in this context.
**John Baez:**Geometric Representation Theory (Lecture 23) (January 29, 2008)- James Dolan on groupoidifying the Hall algebra of the A
_{2}quiver. **Urs Schreiber:**Differential Forms and Smooth Spaces (January 28, 2008)- On turning differential graded-commutative algebras into smooth spaces, and interpreting these as classifying spaces.
**John Baez:**The Yoneda Embedding as a Reflection (January 25, 2008)- A pictorial way to think about the Yoneda embedding (or ‘continuation passing transform’).
**Urs Schreiber:**Integration Without Integration (January 24, 2008)- On how integration and transgression of differential forms is realized in terms of inner homs applied to transport n-functors and their corresponding Lie oo-algebraic connection data.
**John Baez:**Classifying Spaces for 2-Groups (January 24, 2008)- Just as principal bundles are classified by maps into a ‘classifying space’, the same is true for principal 2-bundles.
**John Baez:**Geometric Representation Theory (Lecture 22) (January 18, 2008)- James Dolan groupoidifies the “Hall algebra” of a quiver.
**Urs Schreiber:**Spaceoids (January 17, 2008)- Paolo Bertozzini, Roberto Conti and Wicharn Lewkeeratiyutkul on categorified spaces and the many-object version of C-star algebras.
**David Corfield:**101 things to do with a 2-classifier (January 17, 2008)- What to do with a 2-classifier.
**Urs Schreiber:**Slides:*On the BV-Formalism*(BV Part XI) (January 16, 2008)- A discussion of BV-formalism from a Lie oo-algebraic perspective.
**John Baez:**Strong NDR Pairs — A Technical Question (January 16, 2008)- Some niggling technical questions about topology.
**John Baez:**Geometric Representation Theory (Lecture 21) (January 15, 2008)- The winter quarter of the Geometric Representation Theory seminar begins to tackle examples. Groupoidifying and
*q*-deforming the harmonic oscillator. **Urs Schreiber:**BV-Formalism, Part X: Symplectic Structures (January 14, 2008)- A Lie oo-algebraic perspective on the antibracket.
**David Corfield:**2-Toposes (January 12, 2008)- What happens when you categorify topos theory. You get 2-topos theory… but what’s that like?
**Urs Schreiber:**Ginot and Stiénon on Characteristic Classes of 2-Bundles (January 11, 2008)- Stinot and Stienon on 2-bundles and their characteristic classes.
**John Baez:**The Continuation Passing Transform and the Yoneda Embedding (January 10, 2008)- The Yoneda embedding is familiar in category theory. The continuation passing transform is familiar in computer programming. They’re secretly the same!
**John Baez:**Geometric Representation Theory (Lecture 20) (January 10, 2008)- At last: the Fundamental Theorem of Hecke Operators!
**Urs Schreiber:**The Concept of a Space of States, and the Space of States of the Charged*n*-Particle (January 9, 2008)- On the notion of topos-theoretic quantum state objects, the proposed definition by Isham and Doering and a proposal for a simplified modification for the class of theories given by charged n-particle sigma-models.
**John Baez:**A Tiny Taste of the History of Mechanics (January 7, 2008)- Some incredibly sketchy lecture notes on the history of mechanics from Aristotle to Newton.
**John Baez:**How I Learned to Love the Nerve Construction (January 6, 2008)- Guest post by Tom Leinster on the work of Mark Weber.
**John Baez:**Geometric Representation Theory (Lecture 19) (January 5, 2008)- James Dolan lays the final piece of groundwork for the Fundamental Theorem of Hecke Operators.
**Urs Schreiber:**Dijkgraaf-Witten and its Categorification by Martins and Porter (January 4, 2008)- On Dijkgraaf-Witten theory as a sigma model, and its categorification by Martins and Porter.
**Urs Schreiber:**On BV Quantization, Part IX: Antibracket and BV-Laplacian (January 4, 2008)- The meaning of antibracket, BV-Laplacian and master equation in BV-formalism: Clifford structures on generalized smooth spaces.
**Urs Schreiber:**Comparative Smootheology (January 3, 2008)- A survey and comparison of the various notions of generalized smooth spaces, by Andrew Stacey.
**David Corfield:**Two Cultures in the Philosophy of Mathematics? (January 2, 2008)- Should we have a conference bringing together the two kinds of philosophers of mathematics?
**John Baez:**Geometric Representation Theory (Lecture 18) (January 1, 2008)- An intro to degroupoidification: the process of turning groupoids into vector spaces, and spans of groupoids into linear operators. A key prerequisite: ‘groupoid cardinality’.
**John Baez:**Non-Mathematician Rediscovers R^{n}, C^{n}(December 30, 2007)- Based on the philosophy of Ayn Rand, a nonmathematician has reinvented some familiar algebras — and issued a press release about it.
**Urs Schreiber:**Transgression of*n*-Transport and*n*-Connections (December 30, 2007)- On the general idea of transgression of n-connections and on the underlying machinery of generalized smooth spaces and their differential graded-commutative algebras of differential forms.
**Urs Schreiber:**BF-Theory as a Higher Gauge Theory (December 29, 2007)- On interpreting BF-theory as a higher gauge theory.
**John Baez:**The QG-TQFT Blues (December 29, 2007)- See Scott Carter sing the quantum gravity, topological quantum field theory blues.
**David Corfield:**Challenges for the Future (December 28, 2007)- Benjamin Mann’s 23 challenges.
**John Baez:**Geometric Representation Theory (Lecture 17) (December 26, 2007)- James Dolan on degroupoidification and ‘transfer’ for the zeroth homology of groupoids.
**Urs Schreiber:**Lie oo-Connections and their Application to String- and Chern-Simons*n*-Transport (December 25, 2007)- A discussion of connections for general L-infinity algebras and their application to String- and Chern-Simons n-transport.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 260) (December 25, 2007)- The Retina and Ring Nebulae, my Christmas eve guide to free math and physics books online, and the “exceptional series” of Lie algebras
**John Baez:**Roytenberg on Weak Lie 2-Algebras (December 21, 2007)- Dmitry Roytenberg has proposed a fully general notion of categorified Lie algebra.
**David Corfield:**Progic VI (December 21, 2007)- Graphical models
**Urs Schreiber:**Non-Commutative Structures in Arithmetic and Geometry (December 21, 2007)- A conference on non-commutative structures.
**John Baez:**Geometric Representation Theory (Lecture 16) (December 21, 2007)- Turning a group acting on a set into a groupoid — the ‘weak quotient’ or ‘action groupoid’.
**Urs Schreiber:**Basic Bundle Theory and K-Cohomology Invariants (December 17, 2007)- A book on bundle theory and K-cohomology.
**David Corfield:**Is Mathematics Special? (December 17, 2007)- A conference to be held in Vienna in May 2008 tackles the question “Is Mathematics Special?”
**David Corfield:**A Dialogue on Infinity (December 15, 2007)- The application Alexandre Borovik and I submitted to the John Templeton Foundation as part of their funding of the core theme of infinity was successful. We intend to discuss and disseminate ideas via a blog – A Dialogue on Infinity….
**John Baez:**Geometric Representation Theory (Lecture 15) (December 13, 2007)- James Dolan on various forms of decategorification.
**Urs Schreiber:**Program*Geometry and Physics*at HIM (December 11, 2007)- A program on geometry and physics takes place summer 2008 at the HIM on Bonn.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 259) (December 10, 2007)- Hear what may be lurking in the Egg Nebula, and learn about a mathematical phantom called the “field with one element”.
**Urs Schreiber:**Local Nets from 2-Transport (December 9, 2007)- How to obtain a local net of observables from an extended functorial QFT.
**John Baez:**One Geometry (Math Raps) (December 7, 2007)- A rappin’ proof of the Bolzano–Weierstrass theorem, by Steve Sawin.
**Urs Schreiber:**A Topos for Algebraic Quantum Theory (December 6, 2007)- A summary of some key points of Chris Heunen’s and Bas Spitters’ article.
**David Corfield:**Progic V (December 6, 2007)- I’ve come across something promissing for the Progic project. Apparently there is a way to complete the analogy: propositional logic : predicate logic :: Bayesian networks: ? The answer, it is claimed, is ‘probabilistic relational models’. Now before we…
**Urs Schreiber:**The Principle of General Tovariance (December 5, 2007)- Landsmann proposes that physical laws should be formulated such that they may be internalized into any topos.
**John Baez:**Look Around You (December 3, 2007)- Did you know that the largest number is 45 billion?
**John Baez:**An Unlikely Result (December 3, 2007)- A. N. Other puts ‘An unlikely result’ on the arXiv.
**John Baez:**Astronomers Destroy Universe (December 1, 2007)- Analysis of a media kerfuffle: can looking at the Universe actually hasten its demise? Is that what Krauss and Dent’s paper really said? What
*did*it really say? **Urs Schreiber:**On BV Quantization, Part VIII (November 29, 2007)- Towards understading BV by computing the charged n-particle internal to Z-categories, secretly following AKSZ.
**John Baez:**Geometric Representation Theory (Lecture 14) (November 29, 2007)- Matrix mechanics and Hecke operators. A failed attempt to state the Fundamental Theorem of Hecke Operators.
**John Baez:**Rejecta Mathematica (November 27, 2007)- You can now publish your rejected papers in
*Rejecta Mathematica*. **John Baez:**This Week’s Finds in Mathematical Physics (Week 258) (November 26, 2007)- Read about the flow of grainy substances, polycyclic aromatic hydrocarbons in outer space, and Deligne’s conjecture on Hochschild cohomology.
**John Baez:**Poncelet’s Porism (November 26, 2007)- A question from Gavin Wraith, requiring help from someone who knows elliptic curves and also old-fashioned synthetic geometry.
**John Baez:**Categories, Logic and Physics in London (November 26, 2007)- Bob Coecke and Andreas Döring are running a workshop on ‘Categories, Logic and Foundations of Physics’ at Imperial College.
**Urs Schreiber:**Concordance (November 23, 2007)- On the notion of concordance of 2-bundles and, more generally, on a notion of omega-anafunctor and a possible closed structure on the category of omega-categories with omega-anafunctors between them.
**David Corfield:**MA in Reasoning (November 23, 2007)- I would like to announce that we in the Centre for Reasoning here in Canterbury are launching a new MA course for September 2008. As you can see, this offers the chance to select from four core modules: Logical reasoning,…
**John Baez:**Geometric Representation Theory (Lecture 13) (November 22, 2007)- James Dolan on using braids to describe Hecke operators between flag representations.
**Urs Schreiber:**Something Like Lie-Rinehart ∞-pairs and the BV-complex (BV, part VII) (November 20, 2007)- Notes on something like Lie ∞-algebroids in the light of the BV complex.
**John Baez:**Geometric Representation Theory (Lecture 12) (November 18, 2007)- Heisenberg’s matrix mechanics and its many generalizations, such as the category of relations and the weak 2-category of spans. Understanding Hecke operators in terms of spans.
**John Baez:**Geometric Representation Theory (Lecture 11) (November 16, 2007)- How to describe Hecke operators between flag representations using matrices.
**David Corfield:**Category Theory and Biology (November 15, 2007)- Does category theory have anything to say to biology?
**Urs Schreiber:**Modules for Lie infinity-Algebras (November 13, 2007)- On modules for Lie infinity algebras in general and the definition given by Lars Kjeseth in particular.
**John Baez:**The Dangers of Complex Analysis (November 13, 2007)- Mathematicians like to redefine everyday words and make them into technical terms. This can be dangerous.
**John Baez:**Geometric Representation Theory (Lecture 10) (November 12, 2007)- Simultaneously categorifying and
*q*-deforming Pascal’s triangle will lead us to a categorified quantum group. Here we take the first steps in that direction. **John Baez:**Pfeiffer on Modular Tensor Categories (November 12, 2007)- Hendryk Pfeiffer describes the sort of gadget whose representations form a modular tensor category… and shows how to reconstruct this gadget from its modular tensor category of representations.
**John Baez:**Geometric Representation Theory (Lecture 9) (November 9, 2007)- Getting irreducible representations of symmetric groups from flag representations. Using ‘crackpot matrices’ to describe Hecke operators between flag representations.
**John Baez:**Geometric Representation Theory (Lecture 8) (November 6, 2007)- Categorifying and
*q*-deforming Pascal’s triangle. **Urs Schreiber:**2-Vectors in Trondheim (November 5, 2007)- On line 2-bundles.
**John Baez:**Geometric Representation Theory (Lecture 7) (November 4, 2007)- James Dolan on two applications of Hecke operators: showing that any doubly transitive permutation representation is the direct sum of two irreducible representations, and getting ahold of the irreducible representations of
*n*! **Urs Schreiber:***n*-Bundle Obstructions for Bruce (November 4, 2007)- On the global description of n-bundles obstructing lifts through shifted central extensions.
**Guest:**Concrete Groups and Axiomatic Theories II (November 3, 2007)- More by Todd Trimble on the duality between symmetry and structure: that is, between groups of transformations of finite sets and complete axiomatic theories.
**Urs Schreiber:**On Noether’s Second (BV, Part VI) (October 31, 2007)- On Noether’s second theorem and ghost/antighost pairing.
**Urs Schreiber:**BV for Dummies (Part V) (October 30, 2007)- Some elements of BV formalism, or rather of the Koszul-Tate-Chevalley-Eilenberg resolution, in a simple setup with ideosyncratic remarks on higher vector spaces.
**John Baez:**Comet Holmes (October 30, 2007)- Comet Holmes suddenly got a million times brighter last week, and now it’s visible with the naked eye.
**John Baez:**Higher Clifford Algebras (October 30, 2007)- A talk by Chris Douglas reporting on his work with Arthur Bartels and André Henriques on “higher Clifford algebras”. They’re related to elliptic cohomology and they form a 3-category!
**John Baez:**Fundamental Physics: Where We Stand Today (October 29, 2007)- Slides from an easy talk on the open problems in fundamental physics.
**John Baez:**Steve Fever (October 27, 2007)- A new story by Greg Egan — free online!
**Guest:**Concrete Groups and Axiomatic Theories I (October 26, 2007)- Todd Trimble on the duality between symmetry and structure: that is, between groups of transformations of finite sets and complete axiomatic theories.
**John Baez:**Geometric Representation Theory (Lecture 6) (October 26, 2007)- Categorifying and
*q*-deforming the binomial coefficients. Why are the*q*-binomial coefficients polynomials with natural number coefficients? And, why are they “palindromic” polynomials? Bruhat classes and Schubert cells. **Urs Schreiber:**On String- and Chern-Simons*n*-Transport (October 23, 2007)- Slides on String- and Chern-Simons n-Transport.
**John Baez:**Geometric Representation Theory (Lecture 5) (October 22, 2007)- James Dolan on how to get intertwining operators called “Hecke operators” from relations between types of geometrical figures.
**Urs Schreiber:**On Lie*N*-tegration and Rational Homotopy Theory (October 20, 2007)- On the general ideal of integrating Lie n-algebras in the context of rational homotopy theory, and about Sullivan’s old article on this issue in particular.
**Urs Schreiber:**Advanced Course on Simplicial Methods in Higher Categories (October 19, 2007)- Lectures on simplicial methods in higher category theory.
**John Baez:**Geometric Representation Theory (Lecture 3) (October 19, 2007)- When you have any structure on a set, it has a group of symmetries. Here James Dolan shows how to work backwards: given the symmetries, how read off an axiom system describing the structure those symmetries preserve!
**David Corfield:**Klein 2-Geometry IX (October 18, 2007)- More thoughts on 2-geometry
**Urs Schreiber:**On Weak Cokernels for 2-Groups (October 17, 2007)- On weak cokernels of 2-groups.
**John Baez:**Geometric Representation Theory (Lecture 2) (October 17, 2007)- James Dolan on geometric representation theory: transformation groups, logic and the orbi-simplex.
**Urs Schreiber:***n*-Curvature, Part III (October 16, 2007)- Curvature is the obstruction to flatness. Believe it or not.
**Urs Schreiber:**Loday and Pirashvili on Lie 2-Algebras (secretly) (October 16, 2007)- On the non-standard monoidal structure on 2-term chain complexes induced from the monoidal structure on Baez-Crans 2.vector spaces.
**Urs Schreiber:**What is the Fiber? (October 15, 2007)- On bundle gerbes and whether or not they “are” 2-bundles.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 257) (October 15, 2007)- In “week257”, learn about astrophysics, number theory, topos theory in physics, distributive laws for monads, and hear what’s happening to the Tale of Groupoidification.
**John Baez:**Geometric Representation Theory (Lecture 4) (October 12, 2007)- Categorifying and
*q*-deforming the theory of binomial coefficients — and multinomial coefficients! — using the analogy between projective geometry and set theory. **Urs Schreiber:**Obstructions to*n*-Bundle Lifts Part II: The BIG Diagram (October 12, 2007)- A big diagram illustrating the situation of lifting a bundle through a String-like extension.
**Urs Schreiber:**BV-Formalism, Part IV (October 11, 2007)- Lie algebroids of action groupoids and their relation to BRST formalism.
**Urs Schreiber:**Question on Weak Pullbacks along Sequences (October 11, 2007)- How to weakly pull a functor along a short sequence of groupoids?
**Urs Schreiber:**Categorified Clifford Algebra and weak Lie*n*-Algebras (October 9, 2007)- On weak Lie n-algebras, differential graded Clifford algebra and Roytenberg’s work on weak Lie 2-algebras.
**David Corfield:**Progic IV (October 9, 2007)- More on unity probability theory and logic
**John Baez:**Geometric Representation Theory (Lecture 1) (October 7, 2007)- Video of the first lecture in the new seminar on Geometric Representation Theory at U. C. Riverside.
**Urs Schreiber:**Report on AIM Workshop: Towards Relative Symplectic Field Theory (October 5, 2007)- James Stasheff reports on the AIM workshop Towards Relative Symplectic Field Theory
**Urs Schreiber:**Cohomology of the String Lie 2-Algebra (October 2, 2007)- On the Lie 2-algebra cohomlogy of the String Lie 2-algebra and its relation to twisted K-theory.
**John Baez:**Spans in Quantum Theory (October 1, 2007)- You can use spans to understand why quantum processes act so much like pieces of spacetime.
**David Corfield:**New Blog (September 28, 2007)- Jeff Morton has a new blog!
**David Corfield:**Progic III (September 28, 2007)- More about reconciling logic and probability theory
**Urs Schreiber:**Detecting Higher Order Necklaces (September 27, 2007)- Nils Baas on higher order structures, Enrico Vitale on weak cokernels and a speculation on weak Lie
*n*-algebras triggered by discussion with Pavol Severa. **John Baez:**Rotations in the 7th Dimension (September 26, 2007)- Is there an exceptional isomorphism involving the Spin group in 7 dimensions?
**Urs Schreiber:**An Invitation to Higher Dimensional Mathematics and Physics (September 25, 2007)- A public talk supposed to illustrate some of the basic reasoning underlying higher dimensional algebra.
**David Corfield:**Progic II (September 25, 2007)- More on merging probability theory and logic.
**John Baez:**The Catsters Strike Again: “String Diagrams” (September 24, 2007)- Check out the
*other*Catster. **Urs Schreiber:**Obstructions, Tangent Categories and Lie*N*-tegration (September 24, 2007)- Thoughts on n-bundle theory in terms of Lie n-algebras.
**John Baez:**The Virtues of*American Scientist*(September 24, 2007)- While some science magazines have declined, the
*American Scientist*is still great. **John Baez:**The Catsters’ Latest Hit: “Adjunctions” (September 20, 2007)- Learn about adjunctions on YouTube!
**John Baez:**Deep Beauty: Understanding the Quantum World (September 19, 2007)- A symposium honoring the 75th anniversary of von Neumann’s book on quantum mechanics.
**David Corfield:**Progic (September 18, 2007)- My colleague here in Canterbury Jon Williamson is part of an international research group, progicnet, whose aim is to find a good integration of probability theory and first-order logic. For one reason or another, some technical projects get counted…
**John Baez:**The Catsters on YouTube: “Monads” (September 16, 2007)- Watch some videos from the new group out of Sheffield — the Catsters!
**Urs Schreiber:**Groupoidfest 07 (September 14, 2007)- A conference on groupoids.
**Urs Schreiber:**Obstructions for*n*-Bundle Lifts (September 12, 2007)- On obstructions to lifting the structure n-group of n-bundles.
**David Corfield:**Tim Gowers Joins the Blogosophere (September 11, 2007)- Gowers’ new blog
**Urs Schreiber:**Notes by Jurčo on generalized Bundle Gerbes (September 10, 2007)- Branislav Jurco has some notes on nonabelian bundle 2-gerbes and on associated bundle 1-gerbes.
**Urs Schreiber:**Lie n-Algebra Cohomology (September 7, 2007)- On characteristic classes of n-bundles.
**David Corfield:**Category Theory in Machine Learning (September 5, 2007)- Does category theory have a future in machine learning?
**Urs Schreiber:**Arrow-Theoretic Differential Theory IV: Cotangents (September 3, 2007)- Cotangents and morphisms of Lie n-algebroids from arrow-theoretic differential theory.
**John Baez:**Toward a Higher-Dimensional Wiki (September 2, 2007)- Let’s talk about setting up a wiki for
*n*-category theory and other aspects of higher-dimensional algebra! **John Baez:**Axis of Evil? Or Axis of Opportunity? (August 31, 2007)- A paper by Michael Longo argues that the axes of elliptic galaxies don’t point in random directions.
**Urs Schreiber:**On Hess and Lack on Bundles of Categories (August 30, 2007)- On Hess and Lack’s notion of “bundles of categories”.
**Urs Schreiber:**Question about von Neumann Algebras (August 30, 2007)- On von Neumann algebras and their bimodules.
**John Baez:**Journal Publishers Hire the “Pit Bull of PR” (August 29, 2007)- A publisher’s group called PRISM lobbying against open access journals has hired Eric Dezenhall, the “pit bull of PR”. Learn his strategies.
**Urs Schreiber:**Arrow-Theoretic Differential Theory III: Higher Morphisms (August 28, 2007)- Using arrow-theoretic differential theory to understand higher morphisms of Lie n-algebras.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 256) (August 27, 2007)- Lie
*n*-algebroids and Lie*n*-groupoids in classical mechanics… and the return of the Tale of Groupoidification **David Corfield:**The Reasoner (August 26, 2007)- Contributing to The Reasoner.
**Urs Schreiber:**The G and the B (August 25, 2007)- How to get the bundle governing Generalized Complex Geometry from abstract nonsense and arrow-theoretic differential theory.
**Urs Schreiber:**That Shift in Dimension (August 24, 2007)- What makes the Kontsevich-Cattaneo-Felder theorem tick? How can it be that an n-dimensional quantum field theory is encoded in an (n+1)-dimensional one?
**Urs Schreiber:**Wilson Loop Defects on the String (August 23, 2007)- On Alekseev and Monnier’s work on quantizing Wilson loop observables for the WZW model.
**Urs Schreiber:**Lyakhovich and Sharapov on QFT (On BV-Quantization, Part III) (August 22, 2007)- On Lyakhovich and Sharapov’s insights into mechanics, holography and BV formalism.
**John Baez:**Gerbes in*The Guardian*(August 22, 2007)- A
*Guardian*article about how hard it is to understand the concept of ‘gerbe’. **David Corfield:**Justificatory Narratives (August 22, 2007)- What’s special about special functions.
**Urs Schreiber:**More on Tangent Categories (August 21, 2007)- More comments on the nature of tangent categories and their relation to the notion of shifted tangent bundles to differential graded spaces.
**Urs Schreiber:**On BV Quantization, Part II (August 18, 2007)- A review of elements of the Batalin-Vilkovisky formalism, with an eye towards my claim that this describes configuration spaces which are Lie n-algebroids.
**Urs Schreiber:**On BV Quantization. Part I. (August 17, 2007)- On BV-formalism applied to Chern-Simons theory and its apparent relation to 3-functorial extentended QFT.
**John Baez:**Higher Gauge Theory and the String Group (August 17, 2007)- A talk leading up to the concept of ‘characteristic classes for 2-bundles’, and a nice problem involving them.
**David Corfield:**Future Gazing (August 17, 2007)- What to expect in the next year
**Urs Schreiber:**On Roberts and Ruzzi’s Connections over Posets (August 16, 2007)- On J. Roberts and G. Ruzzi’s concept of G-bundles with connection over posets and its relation to analogous notions discussed at length at the n-Cafe.
**David Corfield:**Math Teach Wiki (August 15, 2007)- Back from Tuscany, I find two e-mails requests awaiting me. First, and I’m now very late on this story, Alexandre Borovik asked me to draw attention to the plight of a Mathematical Summer School held in Turkey. Second, Tim Porter…
**Urs Schreiber:**The Canonical 1-Particle, Part II (August 14, 2007)- More on the canonical quantization of the charged n-particle for the case of a 1-particle propagating on a lattice.
**Urs Schreiber:**Lazaroiu on G-Flows on Categories (August 14, 2007)- Lazaroiu’s concept of a “graded category wiuth shifts” is the same as that of a category “with G-flow”.
**Urs Schreiber:**Question about Tensor Categories (August 13, 2007)- Hendryk Pfeiffer asked me to forward the following question to the Café. Dear nn-category people, I have a question about tensor categories on which I would appreciate comments and references. As probably several people are interested in this, I…
**John Baez:**Math Journal Wiki (August 12, 2007)- A wiki to help us keep an eye on math journals, their prices, their publishers, and their policies.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 255) (August 12, 2007)- In week255, hear what happened at the 2007 Abel Symposium in Oslo. Read explanations of Jacob Lurie and Ulrike Tillman’s talks on cobordism n-categories, Dennis Sullivan and Ralph Cohen’s talks on string topology, Stephan Stolz’s talk on cohomology and…
**Urs Schreiber:**String and Chern-Simons Lie 3-Algebras (August 10, 2007)- A talk on Chern-Simons Lie n-algebras.
**Urs Schreiber:**Arrow-Theoretic Differential Theory, Part II (August 8, 2007)- A remark on maps of categorical vector fields, inner derivations and higher homotopies of L-infinity algebras.
**Urs Schreiber:**Adinkras (August 7, 2007)- The concept of an “Adinkra” - a graph used to describe representations of N-extended d=1 supersymmetry algebras - remarkably resembles some categorical structures which appear in the context of supersymmetry.
**John Baez:**Questions about Modules (August 5, 2007)- What kinds of categories are categories of modules of some ring.
**Urs Schreiber:**Gauge Tranformations of n-Bundles and (n-1)-Gerbes (August 4, 2007)- The notion of gauge transformation of n-bundles and (n-1)-gerbes.
**Urs Schreiber:**QFT of Charged n-Particle: Towards 2-Functorial CFT (August 3, 2007)- Towards a 2-functorial description of 2-dimensional conformal field theory. A project description.
**Urs Schreiber:**QFT of Charged n-Particle: Extended Worldvolumes (August 2, 2007)- Passing from locally to globally refined extended QFTs by means of the adjointness property of the Gray tensor product of the n-particle with the timeline.
**John Baez:**Astronomical London (August 1, 2007)- The Royal Observatory, Greenwich.
**John Baez:**Web Spamming by Academic Publishers (July 31, 2007)- Have you ever thought you were getting a PDF file of a journal article, only to hit a webpage from a publisher demanding money for it? Then you’ve been web spammed.
**John Baez:**Higher Gauge Theory and Elliptic Cohomology (July 31, 2007)- A talk on higher gauge theory and its relation to the work of Stolz and Teichner.
**David Corfield:**Algebra 1*versus*Algebra 2 (July 28, 2007)- Rota on two algebras
**Urs Schreiber:**Arrow-Theoretic Differential Theory (July 27, 2007)- We propose and study a notion of a tangent (n+1)-bundle to an arbitrary n-category. Despite its simplicity, this notion turns out to be useful, as we shall indicate.
**John Baez:**Homotopy Theory and Higher Categories in Barcelona (July 26, 2007)- During the 2007-2008 academic year there will be a program on Homotopy Theory and Higher Categories in Barcelona.
**David Corfield:**Delphic Inspiration (July 25, 2007)- Mathematics and Narrative meeting
**John Baez:**Question About Representations of Finite Groups (July 25, 2007)- If you have a representation of a finite group, can you pick a basis such that all the matrices you get have entries lying in some cyclotomic field?
**Urs Schreiber:**Physical Systems as Topoi, Part III (July 23, 2007)- The third part of the talk.
**Urs Schreiber:**Physical Systems as Topoi, Part II (July 22, 2007)- Second part of the talk.
**Urs Schreiber:**Physical Systems as Topoi, Part I (July 22, 2007)- Doering on his work with Isham on topos theoretic models of quantum theory and general theories of physics.
**Urs Schreiber:**Making AdS/CFT Precise (July 22, 2007)- The last session of Recent Developments in QFT in Leipzig was general discussion, which happened to be quite interesting for various reasons….
**Urs Schreiber:**Deformation Quantization of Surjective Submersions (July 22, 2007)- Waldman on noncommutative deformations of total spaces of fiber bundles.
**Urs Schreiber:**Tangent Categories (July 21, 2007)- An arrow-theoretic formulation of tangency and its relation to inner automorphisms.
**John Baez:**Astronomical Paris (July 18, 2007)- A tour of L’Observatoire de Paris.
**David Corfield:**Mathematical Imperatives (July 16, 2007)- Mathematical ‘must’
**David Corfield:**George Mackey (July 14, 2007)- Who will collect the letters of George Mackey?
**John Baez:**This Week’s Finds in Mathematical Physics (Week 254) (July 13, 2007)- Connes and Marcolli’s new book, Witten’s new paper, exceptional Lie superalgebras and the Standard Model… and the Tale of Groupoidification, continued.
**David Corfield:**Breaking Out of the Box (July 13, 2007)- Try n-categories in spheres!
**David Corfield:**Return of the Euler Characteristic of a Category (July 9, 2007)- Leinster’s second paper on the Euler characteristic.
**John Baez:**Derek Wise on Cartan Geometry and MacDowell–Mansouri Gravity (July 7, 2007)- Derek Wise’s thesis: how the geometry of the MacDowell–Mansouri and Chern–Simons approaches to gravity become clearer if we use Cartan connections.
**John Baez:**Mathematical Paris (July 7, 2007)- In Paris, you can get your name on a street sign just by being good in math.
*Really*good. **David Corfield:**Kernels in Machine Learning III (July 6, 2007)- More about kernels, including heat kernels, in machine learning.
**Urs Schreiber:**Multiplicative Structure of Transgressed n-Bundles (July 5, 2007)- On how to think of multiplicative structures on n-bundles with connection.
**Urs Schreiber:**Supercategories (July 4, 2007)- A proposal for a definition of “supercategory” and an arrow-theoretic understanding of some elements of supersymmetry.
**Urs Schreiber:**The Inner Automorphism 3-Group of a Strict 2-Group (July 4, 2007)- On the definition and construction of the inner automorphism 3-group of any strict 2-group, and how it plays the role of the universal 2-bundle.
**David Corfield:**Kernels in Machine Learning II (June 28, 2007)- More about kernels in machine learning.
**Urs Schreiber:**Conference on Categories in Geometry and Physics (June 28, 2007)- A conference on categories and their application in geometry and physics, taking place in Split, Croatia.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 253) (June 28, 2007)- From the Standard Model to SU(5), SO(10), E
_{6}… and maybe even on to E_{8}, with a friendly tip of the hat to symmetric spaces like the complexified octonionic plane. **Urs Schreiber:**Colorings of Graphs (June 26, 2007)- Dmitry Kozlov on understanding the graph coloring problem by looking at simplicial Hom-complexes of graphs.
**David Corfield:**Kernels in Machine Learning I (June 25, 2007)- The use of kernels in machine learning
**Urs Schreiber:**Some Recreational Thoughts on Super-Riemannian Cobordisms (June 23, 2007)- On super-Riemannian structure on cobordisms in terms of super-Poincaré connections.
**David Corfield:**Faith and Reason (June 21, 2007)- Polanyi on faith and reason
**Urs Schreiber:**Curvature, the Atiyah Sequence and Inner Automorphisms (June 20, 2007)- On the notion of curvature 2-functor in light of morphisms from the path sequence of the base to the Atiyah sequence of the bundle.
**David Corfield:**Degeneracy (June 19, 2007)- Papers on degenerate n-categories
**John Baez:**Cohomology and Computation (Week 27) (June 19, 2007)- Defining the cohomology of algebraic gadgets using the bar construction.
**Urs Schreiber:**Waldorf on Parallel Transport Functors (June 18, 2007)- Review of parallel transport functors by Konrad Waldorf.
**Urs Schreiber:**Generalized Geometric Langlands is False (June 18, 2007)- C. Teleman on a counter example to the generalised geometric Langlands conjecture.
**John Baez:**Cohomology and Computation (Week 26) (June 18, 2007)- An example of the bar construction: puffing up a point to the free contractible
*G*-space*EG*, important in group cohomology. **John Baez:**More Mysteries of the Number 24 (June 17, 2007)- Kummer’s 24 hypergeometric functions: what are they?
**David Corfield:**Opetopes as Trees (June 16, 2007)- A new paper on opetopes.
**Urs Schreiber:**Polyvector Super-Poincaré Algebras (June 14, 2007)- Superextension of Poincare algebras and how these give rise to brane charges.
**John Baez:**Why Math Teachers Get Grumpy (June 13, 2007)- Grading final exams — a test of ones soul.
**Urs Schreiber:**Spectral Triples and Graph Field Theory (June 12, 2007)- Yan Soibelman is thinking about spectral stringy geometry.
**David Corfield:**Two ArXiv Papers (June 12, 2007)- Two papers on bicategories.
**David Corfield:**More Mathematical Blogging (June 12, 2007)- A new mathematics blog.
**Urs Schreiber:**Extended QFT and Cohomology II: Sections, States, Twists and Holography (June 10, 2007)- How transformations of extended d-dimensional quantum field theories are related to (d-1)-dimensional quantum field theories. How this is known either as twisting or as, in fact, holography.
**John Baez:**Quantization and Cohomology (Week 27) (June 8, 2007)- Review of what we’ve done in this course; prospectus of what’s still to be done.
**Urs Schreiber:**Extended Quantum Field Theory and Cohomology, I (June 8, 2007)- On understanding extended quantum field theory and generalized cohomology.
**Urs Schreiber:**Large Smooth Categories (June 8, 2007)- Stacks versus categories internal to sheaves.
**Urs Schreiber:**The n-Café Quantum Conjecture (June 8, 2007)- Why it seems that quantum mechanics ought to be the de-refinement of a refined theory which lives in one categorical degree higher than usual.
**David Corfield:**The Curious Incident of the Dog in the Night-time (June 8, 2007)- The cyclic category.
**David Corfield:**Categorifying Quantum Mechanics (June 7, 2007)- Two approaches to categorifying quantum mechanics — compare them!
**John Baez:**Cohomology and Computation (Week 25) (June 6, 2007)- Getting monads, comonads and simplicial objects from adjoint functors.
**David Corfield:**Whose 2-Vector Spaces? (June 6, 2007)- 2-vector spaces for elliptic cohomology.
**David Corfield:**June Events (June 5, 2007)- June conferences
**John Baez:**Quadratic Reciprocity (June 4, 2007)- Quadratic Reciprocity is one of the gems of elementary number theory. Why does this famous proof actually work?
**Urs Schreiber:**Connections on String-2-Bundles (June 3, 2007)- On connections on String 2-bundles.
**John Baez:**Quantization and Cohomology (Week 26) (June 1, 2007)- Cech cohomology in terms of anafunctors and ananatural transformations.
**David Corfield:**The Woodstock of the Mind (June 1, 2007)- The Hay Festival
**John Baez:**Quantization and Cohomology (Week 25) (June 1, 2007)- How describing bundles in terms of Cech cohomology secretly amounts to describing them in terms of smooth anafunctors.
**John Baez:**Quantization and Cohomology (Week 24) (June 1, 2007)- Three approaches to connections and gauge transformations, leading up to one based on Toby Bartels’ notion of “smooth anafunctor”.
**Urs Schreiber:**On the Bar Construction (May 31, 2007)- Todd Trimble on the bar construction.
**Urs Schreiber:**The Second Edge of the Cube (May 31, 2007)- Differentiating parallel transport anafunctors to Cartan-Ehresmann connections.
**Urs Schreiber:**Kock on Higher Connections (May 31, 2007)- Anders Kock gives a synthetic differential description of parallel n-transport using strict n-fold categories.
**Urs Schreiber:**What is a Lie Derivative, really? (May 31, 2007)- On the arrow-theory behind Lie derivatives.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 252) (May 28, 2007)- The long-range weather report on Neptune, hot Neptunes in other solar systems, the electromagnetic snake at the center of our galaxy, and Hecke operators.
**John Baez:**Link Homology and Categorification in Kyoto (May 26, 2007)- Link Homology and Categorification — a conference in Kyoto, May 10-25, 2007.
**John Baez:**Congratulations to David! (May 25, 2007)- David Corfield found a permanent position at the University of Kent at Canterbury!
**Urs Schreiber:**Derivation Lie 1-Algebras of Lie n-Algebras (May 25, 2007)- On ordinary Lie algebras “of derivations” of Lie n-algebras.
**David Corfield:**Going Hi-Tech (May 25, 2007)- Slides on n-categories
**John Baez:**Cohomology and Computation (Week 24) (May 24, 2007)- What makes the bar construction tick?
**David Corfield:**The Two Cultures of Mathematics Revisited (May 22, 2007)- Revisiting the idea of two cultures in mathematics.
**John Baez:**Questions on n-Categories and Topology (May 21, 2007)- Questions by Bruce Westbury on $n$-categories and topology
**Urs Schreiber:**Chern Lie (2*n*+1)-Algebras (May 19, 2007)- Generalizing Baez-Crans Lie 2n-algebras to Chern and to Chern-Simons Lie (2n+1)-algebras.
**Urs Schreiber:**Calculations Inside Semisimple Categories (May 18, 2007)- Bruce Bartlett on computations in semisimple categories.
**David Corfield:***Linear Algebra Done Right*(May 18, 2007)- Doing linear algebra right! — the trouble with determinants.
**John Baez:**Penrose on Angular Momentum: An Approach to Combinatorial Space-Time (May 18, 2007)- Roger Penrose’s paper Angular Momentum: an Approach to Combinatorial Space-Time is now available in electronic form.
**David Corfield:**Two Physics Papers Involving Categorification (May 16, 2007)- Physics papers dealing with categorification
**John Baez:**People Who May or May Not Actually Exist (May 12, 2007)- A lot of famous people may or may not actually exist. What are the odds?
**John Baez:**Puzzle #12 (May 12, 2007)- What’s the size of New Jersey, shaped like a dog bone, and named after a famous woman?
**John Baez:**Cohomology and Computation (Week 23) (May 11, 2007)- The bar construction: a way to get simplicial sets from algebraic gadgets.
**Urs Schreiber:**Zoo of Lie n-Algebras (May 10, 2007)- A menagerie of examples of Lie n-algebras and of connections taking values in these, including the String 2-connection and the Chern-Simons 3-connection.
**John Baez:**Quantization and Cohomology (Week 23) (May 10, 2007)- Connections on principal bundles — a modern treatment.
**Urs Schreiber:**Workshop on Elliptic Cohomology in Hamburg (May 9, 2007)- A little workshop on elliptic cohomology, mainly the functorial QFT approach by Stolz and Teichner, in Hamburg, June 2007.
**Urs Schreiber:**Report from “Workshop on Higher Gauge Theory” (May 9, 2007)- Report-back on a little symposium titled “Higher Gauge Theory” (but concerned just with abelian gerbes) that took place at the AEI in Golm.
**John Baez:**Cohomology and Computation (Week 22) (May 8, 2007)- How to count holes.
**John Baez:**Quantization and Cohomology (Week 22) (May 8, 2007)- Smooth functors and beyond: why we need nontrivial principal U(1) bundles to describe phases in geometric quantization.
**John Baez:**Support the Banff Protocol! (May 8, 2007)- There’s a new AMS Notices article on how the board of
*Topology*resigned to protest Elsevier’s high prices. Support the Banff Protocol — avoid publishing in highly expensive journals! **David Corfield:**Klein 2-Geometry VIIIS (May 7, 2007)- A few additional Klein 2-geometry thoughts.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 251) (May 6, 2007)- Foundations of quantum mechanics — the news from Les Treilles.
**Urs Schreiber:**The First Edge of the Cube (May 4, 2007)- The notion of smooth local i-trivialization of transport n-functors for n=1.
**Urs Schreiber:**Grandis on Collared Cobordisms and TQFT (May 2, 2007)- Marco Grandis discusses the arrow-theory of collared cobordisms and topological quantum field theory.
**Urs Schreiber:**Symposium on “Higher Gauge Theory” in Golm (May 2, 2007)- A workshop on abelian gerbes, Deligne cohomology and strings.
**David Corfield:**Probability in Amsterdam (May 1, 2007)- Foundations of the Formal Sciences conference in Amsterdam.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 250) (April 26, 2007)- The Popescu-Rohrlich game, double cosets, atomic invariant relations and spans of groupoids.
**Urs Schreiber:**n-Curvature (April 25, 2007)- The n-curvature canonically associated with a transport n-functor.
**David Corfield:**Learning from Our Ancestors (April 25, 2007)- What can we learn from earlier mathematicians?
**David Corfield:**Who’s on the Right Track? (April 23, 2007)- Connes on the wrong track for quantum gravity.
**Urs Schreiber:**Report-Back on BMC (April 22, 2007)- Bruce Bartlett reports from the British Mathematics Colloquium 2007.
**John Baez:**Cohomology and Computation (Week 21) (April 20, 2007)- Why mathematicians like to take algebraic gadgets and topological spaces and turn them into simplicial sets.
**John Baez:**Quantization and Cohomology (Week 21) (April 20, 2007)- Understanding the ‘action’ in classical mechanics as a smooth functor — the case of a path groupoid.
**John Baez:**Cohomology and Computation (Week 20) (April 20, 2007)- Why do mathematicians like simplices so much? Why are they better than other shapes?
**Urs Schreiber:**Some Conferences (April 19, 2007)- A conference on bundles and gerbes, another one on topology, and comments on associated 2-vector bundles and String connections.
**David Corfield:**The Field With One Element (April 17, 2007)- For some grand theory building and an answer to the question ‘What is the field with one element?’, see Nikolai Durov’s New Approach to Arakelov Geometry.
**David Corfield:**The Two Cultures of Mathematics (April 17, 2007)- ‘Theory-builders’ and ‘problem-solvers’.
**John Baez:***Incandescence*(April 14, 2007)- Greg Egan’s next novel is coming out in 2008 — but you can already read a story set in the same universe.
**John Baez:**Topos Theory in the*New Scientist*(April 14, 2007)- Robert Matthews on Chris Isham’s work on topos theory and physics.
**John Baez:**Schur Functors (April 12, 2007)- What’s the right way to think about Schur functors?
**David Corfield:**Structure and Pseudorandomness (April 12, 2007)- Terence Tao has written three delightful posts detailing his views on the relationship between structure and pseudorandomness in mathematics.
**David Corfield:**Category Theory as Esperanto (April 11, 2007)- If category theory is the Esperanto of mathematics, how many mathematicians speak it fluently?
**John Baez:**Quantization and Cohomology (Week 20) (April 11, 2007)- Generalizing smooth manifolds to smooth spaces, so we can define smooth categories and study the principal of least action in a very general setting.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 249) (April 9, 2007)- Klein’s Erlangen program, now available in English! And: the Tale of Groupoidification, continued.
**John Baez:**Why Mathematics Is Boring (April 6, 2007)- Storytellers have developed many strategies for luring in readers and keeping them interested. Mathematicians systematically avoid these.
**David Corfield:**Whatever Happened to the Categories? (April 6, 2007)- Are we doing our job as broadcasters well? Max Tegmark has a new paper out on the physical universe as an abstract mathematical structure. Not a whiff of categories, let alone nn-categories. Tegmark has read some of philosophy of…
**John Baez:**Cohomology and Computation (Week 19) (April 6, 2007)- The origins of cohomology in the study of ‘syzygies’, or relations between relations.
**David Corfield:**Automated Theorem Proving (April 5, 2007)- In Brussels, we heard from Koen Vervloesem about attempts towards better automated theorem provers. Readers of my book will know that I devoted its second chapter to automated theorem provers, to provide a relief against which to consider ‘real…
**John Baez:**Quantization and Cohomology (Week 19) (April 4, 2007)- Finding critical points of the action in a general context: a smooth category equipped with a smooth ‘action’ functor.
**Urs Schreiber:**Oberwolfach CFT, Tuesday Morning (April 3, 2007)- On Q-systems, on the Drinfeld Double and its modular tensor representation category, and on John Roberts ideas on nonabelian cohomology and QFT.
**Urs Schreiber:**Oberwolfach CFT, Monday Evening (April 3, 2007)- Some random notes concerning, bundles, functorial quantum field theory and algebraic QFT.
**David Corfield:**Rota on Combinatorics (April 2, 2007)- From an interview with Gian-Carlo Rota and David Sharp: Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that…
**Urs Schreiber:**Oberwolfach CFT, Monday Morning (April 2, 2007)- Some basic concepts and facts concerning local representations of nets of local algebras on the real line.
**Urs Schreiber:**Oberwolfach CFT, Arrival Night (April 1, 2007)- Some musings on the relation of AQFT to functorial QFT.
**David Corfield:**Bernard Williams on Scientism (April 1, 2007)- In Brussels, Brendan Larvor took us through a range of options for those of us who want our philosophy of mathematics to take serious notice of the history of mathematics. A distinction he relied upon was one Bernard Williams introduced…
**Urs Schreiber:**Some Notes on Local QFT (March 31, 2007)- Some aspects of the AQFT description of 2d CFT.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 248) (March 29, 2007)- Watch movies of coronal mass ejections, auroras, and tornados on the Sun! Then: the Tale of Groupoidification continues.
**John Baez:**Relativity on the World-Wide Web (March 28, 2007)- Chris Hillman is back with a new improved guide to learning relativity!
**David Corfield:**Philosophising in Brussels (March 24, 2007)- I’m off to Brussels tomorrow to speak at a conference called Perspectives on Mathematical Practices 2007. You can see the notes for my talk.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 247) (March 23, 2007)- Symmetry: from the ancient Greeks through medieval Islam to E
_{8}and groupoidification. **Urs Schreiber:**History of Understanding Bundles with Connection using Parallel Transport around Loops (March 23, 2007)- A list of some papers involved in the historical development of the idea of expressing bundles with connection in terms of their parallel transport around loops.
**Urs Schreiber:**Branes, Bi-Branes, 2-Vectors, 2-Linear Maps (March 22, 2007)- In which sense D-branes generalize to “bi-branes”.
**Urs Schreiber:**Recent Developments in QFT in Leipzig (March 21, 2007)- A conference on new developments in Quantum Field Theory at the Max Planck Institute in Leipzig.
**David Corfield:**Generalising Hopf Algebras (March 20, 2007)- Gizem Karaali’s paper ‘On Hopf algebras and their generalizations’, in which she describes Hopf algebras and five attempts to generalise them.
**Urs Schreiber:**QFT of Charged n-Particle: The Canonical 1-Particle (March 19, 2007)- On the category of paths whose canonical Leinster measure reproduces the path integral measure appearing in the quantization of the charged particle.
**John Baez:**News about E_{8}(March 19, 2007)- A huge new calculation involving the exceptional Lie group E
_{8}. **John Baez:**Quantum Logic, Topology and Categories at Oxford (March 16, 2007)- Conferences on categories, logic, computation and topology at Oxford this summer.
**Urs Schreiber:**Poisson-Sigma Models, Lie Algebroids, Deformations and Higher Analogues in Vienna (March 16, 2007)- Announcement of ESI Workshop on Lie Algebroids in Summer 2007.
**John Baez:**Classical vs Quantum Computation (Week 18) (March 15, 2007)- 2-categories of computation, and an intriguing relation to catastrophe theory.
**Urs Schreiber:**Star-Structures and Daggers (March 15, 2007)- Question on the compatibility of daggers and stars in monoidal 2-categories with duals.
**David Corfield:**Manin on Mathematics (March 15, 2007)- On the ArXiv today, Yuri Manin has one of those wide-ranging overviews of the life of mathematics: Mathematical knowledge: internal, social and cultural aspects. One comment - When Poincaré said that there are no solved problems, there are only problems…
**Urs Schreiber:**Differentiating Lie Groupoids to Lie Algebroids (March 14, 2007)- On an attempt to find a nice way to formulate the definition of the Lie algebroid obtained from a Lie groupoid.
**John Baez:**Quantization and Cohomology (Week 18) (March 14, 2007)- From particles to strings. First: building a Hilbert space from a category C equipped with an "amplitude" functor A: C → U(1). Then: building a 2-Hilbert space from a 2-category C equipped with a 2-functor A: C → U(1)Tor.
**John Baez:**Snowglobe Models (March 10, 2007)- Constructing strange ‘nonstandard models’ of typed λ-calculi.
**John Baez:**We’re Under Attack! (March 9, 2007)- We’re experiencing problems due to a trackback spambot.
**John Baez:**Gurski on Tricategories (March 9, 2007)- If your work takes you into the land of weak 3-categories, you’ll need to read this.
**John Baez:**Classical vs Quantum Computation (Week 17) (March 9, 2007)- String worldsheets illustrating processes of computation!
**Urs Schreiber:**Canonical Measures on Configuration Spaces (March 8, 2007)- On how the Leinster weighting on a category might provide path integral measures in physics.
**John Baez:**Quantization and Cohomology (Week 17) (March 7, 2007)- Getting Hilbert spaces and operator algebras from categories.
**Urs Schreiber:**QFT of Charged n-Particle: Sheaves of Observables (March 6, 2007)- On the concepts of sheaves and nets of algebras of observables in quantum field theory.
**David Corfield:**Computer Science and Physics (March 5, 2007)- Samson Abramsky’s paper:
*What are the fundamental structures of concurrency? We still don’t know!* **Urs Schreiber:**QFT of Charged n-Particle: Disk Path Integral for String in trivial KR Field (March 5, 2007)- The arrow-theoretic perspective on the path integral for the disk diagram of the open string.
**John Baez:**A Topos Foundation for Theories of Physics (March 3, 2007)- A major new paper by Isham and Döring.
**David Corfield:**Dynamics of Mathematical Reason (March 2, 2007)- I’m having a spot of bother getting a paper published. It’s about the philosopher Michael Friedman’s treatment of mathematics in his
*Dynamics of Reason*. I’d be grateful for any comments from the Café clientele. **John Baez:**Classical vs Quantum Computation (Week 16) (March 2, 2007)- The λ-calculus for… high school calculus!
**Urs Schreiber:**QFT of Charged n-Particle: Gauge Theory Kinematics (February 28, 2007)- A review of Fleischhack’s discussion of a Weyl algebra on spaces of generalized connections.
**Urs Schreiber:**QFT of Charged n-Particle: Algebra of Observables (February 28, 2007)- The algebra of observables as certain endomorphisms of the n-category of sections.
**John Baez:**Quantization and Cohomology (Week 16) (February 27, 2007)- More examples of path-integral quantization: the particle in a potential.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 246) (February 25, 2007)- The trouble with fundamental physics today.
**Urs Schreiber:**Amplimorphisms and Quantum Symmetry, II (February 25, 2007)- A remark on the appearance of iterated module
*n*-categories in quantum field theory. **David Corfield:**The Health Book (February 23, 2007)- The book —
*Why Do People Get Ill?*— is finally out there in the shops. **John Baez:**How to Write Mathematics Badly (February 23, 2007)- A hilarious video.
**John Baez:**Classical vs Quantum Computation (Week 15) (February 23, 2007)- The λ-theory of commutative rings.
**David Corfield:**Noncommutative Geometry Blog (February 21, 2007)- A new blog on noncommutative geometry.
**David Corfield:**Cake Talk (February 21, 2007)- A talk on Bayesianism, information geometry, and nonparametric statistics.
**John Baez:**An Introduction to Algebraic Topology (February 21, 2007)- Want to learn a little algebraic topology?
**John Baez:**Quantization and Cohomology (Week 15) (February 21, 2007)- An example of path integral quantization: the free particle on a line (continued).
**David Corfield:**Congratulations! (February 17, 2007)- It’s John’s wedding day today!
**John Baez:**Classical vs Quantum Computation (Week 14) (February 16, 2007)- From typed lambda-calculus to cartesian closed categories and back.
**Urs Schreiber:**QFT of Charged n-Particle: T-Duality (February 16, 2007)- Topological T-duality as a pull-push transformation of sections of the 2-particle.
**John Baez:**Quantization and Cohomology (Week 14) (February 15, 2007)- An example of path integral quantization: the free particle on a line.
**Urs Schreiber:**Higher Morphisms of Lie n-Algebras and L-infinity Algebras (February 15, 2007)- On higher morphisms of Lie n-algebras, and higher homotopies between L_oo algebras.
**David Corfield:**Category Theoretic Probability Theory II (February 13, 2007)- More on what category theory has to say about probability theory.
**Urs Schreiber:**Slides for Freed’s Andrejewski Lecture (February 13, 2007)- Lecture by Freed on Freed-Hopkins-Teleman.
**Urs Schreiber:**QFT of Charged n-Particle: Dynamics (February 12, 2007)- Definition of the dynamics of the charged quantum particle by pull-push along correspondences of path spaces.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 245) (February 12, 2007)- Read about Fields Institute workshop on Higher Categories and Their Applications.
**David Corfield:**Why Do I Bother? (February 10, 2007)- Terence Tao’s paper ‘What is good mathematics?’, and what the philosophy of mathematics might say about this.
**John Baez:**Classical vs Quantum Computation (Week 13) (February 9, 2007)- Lambek and Scott’s definition of a typed lambda-calculus.
**Urs Schreiber:**Day on RCFTs (February 9, 2007)- Brian Day on a relation between fusion rings, braided tensor categories and “realisation” in enriched category theory.
**John Baez:**Infinitely Categorified Calculus (February 9, 2007)- In noncommutative geometry, we must infinitely categorify the calculus!
**Urs Schreiber:**Isham on Arrow Fields (February 8, 2007)- On Chris Isham’s notion of an “arrow field” on a category.
**Urs Schreiber:**QFT of Charged n-particle: Chan-Paton Bundles (February 7, 2007)- Chan-Paton bundles from the pull-push quantization of the open 2-particle.
**David Corfield:**Category Theoretic Probability Theory (February 7, 2007)- Having noticed (e.g., here and here) that what I do in my day job (statistical learning theory) has much to do with my hobby (things discussed here), I ought to be thinking about probability theory in category theoretic terms….
**John Baez:**Quantization and Cohomology (Week 13) (February 6, 2007)- Statistical mechanics and temperature-dependent mathematics.
**Urs Schreiber:**Amplimorphisms and Quantum Symmetry, I (February 6, 2007)- Localized endomorphisms of quantum observables in arrow-theoretic terms.
**Urs Schreiber:**Amplimorphisms (February 5, 2007)- A certain notion of twisted morphisms of algebras – and how to think about it.
**David Corfield:**In the Footsteps of Rudolf Carnap II (February 5, 2007)- The next day I set off East to Jena, following the path taken by Carnap, and by my host, David Green, a British mathematician who works on the cohomology of finite groups. While in Wuppertal, David had become interested…
**David Corfield:**In the Footsteps of Rudolf Carnap I (February 5, 2007)- Last week I gave a couple of talks in Germany. Thursday saw me in the town of Wuppertal, famous for its Schwebebahn, a railway built above the river Wupper, which snakes its way through the middle of the town. As…
**John Baez:**This Week’s Finds in Mathematical Physics (Week 244) (February 3, 2007)- How to measure the size of a category.
**Urs Schreiber:**Higher Structure in Geometry and Physics – and in Paris (February 2, 2007)- Slides of talks for a conference on higher structures in geometry and physics
**Urs Schreiber:**Huisken on Uniformization, II (February 2, 2007)- Something about Ricci flow and the proof of the Poincare conjecture.
**Urs Schreiber:**Huisken on Uniformization, I (February 2, 2007)- Some ideas behind the proof of the Poincare conjecture by means of Ricci flow.
**John Baez:**Classical vs Quantum Computation (Week 12) (February 1, 2007)- How to see computation as a process: rewrite rules.
**Urs Schreiber:**Towards the FFRS Description of 2dCFT (B) (February 1, 2007)- “2-processes”: 2-morphisms, adjunctions, the exchange law and the Frobenius property
**John Baez:**In Memory of Max Kelly (February 1, 2007)- Letters in appreciation of Max Kelly.
**John Baez:**Quantization and Cohomology (Week 12) (January 31, 2007)- Classical mechanics, quantum mechanics and statistical mechanics as ‘matrix mechanics’ over various rigs (ri
**n**gs without**n**negatives). **David Corfield:**No Need to Apologise (January 29, 2007)- Café regular John Armstrong has a blog. It goes by the name of The Unapologetic Mathematician. A subtle allusion to Hardy’s A Mathematician’s Apology, playing cleverly on the two meanings of apology?…
**Urs Schreiber:**CFT in Oberwolfach (January 29, 2007)- An Oberwolfach meeting on conformal field theory.
**David Corfield:**Another Interview (January 28, 2007)- It’s worth taking a look at an interview Mikio Sato gave to Emmanuel Andronikof in 1990, published in February’s Notices of the American Mathematical Society. Sato is famous for algebraic analysis, D-modules, and the like, about which I know next…
**David Corfield:**Peering Through the Veil (January 27, 2007)- Twice in recent days I have confronted the possibility of experiencing a kind of alienation due to interviews. First, my co-author Darian Leader and I were interviewed by the New Scientist about our book Why Do People Get Ill?. A…
**Urs Schreiber:**Globular Extended QFT of the Charged n-Particle: String on BG (January 26, 2007)- The string on the classifying space of a strict 2-group.
**John Baez:**Classical vs Quantum Computation (Week 11) (January 25, 2007)- Lots of examples of 2-categories!
**John Baez:**Classical vs Quantum Computation (Week 10) (January 24, 2007)- Categorifying the concept of ‘category’!
**Urs Schreiber:**The Globular Extended QFT of the Charged n-Particle: Definition (January 24, 2007)- Turning a classical parallel transport functor on target space into a quantum propagation functor on parameter space.
**Urs Schreiber:**Cocycle Category (January 24, 2007)- Bruce Bartlett reports on Rick Jardine’s concept of ‘cocycle categories’ and their relation to anafunctors.
**John Baez:**Quantization and Cohomology (Week 11) (January 24, 2007)- What’s really going on with quantization?
**David Corfield:**Two Café Owners Interviewed (January 23, 2007)- As neither John nor Urs has announced it, readers might like to find out about their motivations for starting and running this blog in an interview they gave to Bruce Bartlett, available in written form and also as an MP3…
**John Baez:**Higher Categories and Their Applications (January 23, 2007)- The Fields Institute workshop on
*n*-categories was a lot of fun. If you couldn’t make it, you can still see what it was like. **Urs Schreiber:**Fusion and String Field Star Product (January 21, 2007)- A description of the string field star product and of fusion of loop group representations from the point of view of functorial transport.
**John Baez:**Animals That Are Also Verbs (January 21, 2007)- A fly can fly. A duck can duck. And a dog can dog you.
**David Corfield:**More on Duality (January 19, 2007)- Continuing our earlier discussion about duality, it’s worth noting a distinction that Lawvere and Rosebrugh introduce in chapter 7 of their Sets for Mathematics between ‘formal’ and ‘concrete’ duality. Formal duality concerns mere arrow reversal in the relevant diagrams,…
**John Baez:**Traces in Ottawa (January 19, 2007)- A workshop on applications of traces to algebra, analysis and categorical logic.
**John Baez:**Knot Homology in Faro (January 19, 2007)- This year’s Oporto Meeting Geometry, Topology and Physics will actually be held in Faro, The theme is knot homology!
**John Baez:**Multiplicative Intuitionistic Linear Logic (January 18, 2007)- Translating from how logicians think about mulitplicative intuitionistic linear logic to how category theorists might think about it.
**David Corfield:**Duality between Probability and Optimization (January 18, 2007)- One of the reasons I have an interest in what we find out about mechanics in different rigs is that many machine learning algorithms are expressible in thermodynamic form, as the tutorial, Energy-Based Models: Structured Learning Beyond Likelihoods, by…
**Urs Schreiber:**D-Branes from Tin Cans, III: Homs of Homs (January 18, 2007)- Sections of sections, their pairing and n-disk correlators.
**John Baez:**Quantization and Cohomology (Week 10) (January 17, 2007)- Classical versus quantum mechanics: the Lagrangian and Hamiltonian approaches
**Urs Schreiber:**Towards the FFRS Description of 2dCFT (A) (January 16, 2007)- 2-dimensional cobordisms, topological quantum field theory and Frobenius algebras
**Urs Schreiber:**Aaronson on the Nature of Quantum Mechanics (January 16, 2007)- Scott Aaronson on the nature of quantum mechanics.
**Urs Schreiber:**Khovanov Homology (January 14, 2007)- Khovanov homology and its generalization from links to tangles.
**Urs Schreiber:**The First Part of the Story of Quantizing by Pushing to a Point… (January 13, 2007)- Taking sections of transport functors by pushing them to a point.
**David Corfield:**Ubiquitous Duality (January 11, 2007)- I’m in one of those phases where everywhere I look I see the same thing. It’s Fourier duality and its cousins, a family which crops up here with amazing regularity. Back in August, John wrote: So, amazingly enough, Fourier duality…
**David Corfield:**Universality in Particularity (January 8, 2007)- To keep me from brooding on the pleasure I’m missing out on by not being with my Café co-hosts in Toronto, let me try out a blog post. In just about every academic endeavour to which I’ve applied myself, I…
**John Baez:**Classical vs Quantum Computation (Week 9) (January 5, 2007)- Now we’ll categorify the concept of ‘category’ to see computation as a process!
**Urs Schreiber:**FFRS on Uniqueness of CFT: Sewing as Natural Transformation (January 4, 2007)- On FFRS’s formulation of sewing constraints in terms of natural transformations.
**Urs Schreiber:**FFRS on Uniqueness of CFT: Morphisms into Transport Functors (January 3, 2007)- On morphisms into functors and states and sections in the FFRS description of conformal field theory.
**Urs Schreiber:**FFRS on Uniqueness of Conformal Field Theory (January 3, 2007)- A strengthening of the FFRS theorem on 2-dimensional rational conformal field theory.
**David Corfield:**Research Proposals (January 3, 2007)- There’s nothing quite like a research proposal to give you a sense of some of the big stories out there. Try Geometry and Quantum Theory for what’s happening in Holland of relevance to the Café. From a couple of years…
**John Baez:**Puzzle #11 (December 31, 2006)- When was the toothpick invented?
**John Baez:**This Week’s Finds in Mathematical Physics (Week 243) (December 26, 2006)- In week243 of This Week’s Finds, hear about Claude Shannon, his sidekick Kelly, and how they used information theory to make money at casinos and the stock market. Hear about the new book Fearless Symmetry, which explains fancy number…
**John Baez:**The Earliest Stars? (December 25, 2006)- Have people caught sight of the earliest stars in the Universe?
**John Baez:**arXiv Policy Statement? (December 24, 2006)- Where is the arXiv’s official policy statement about guaranteed open access?
**John Baez:**Puzzle #9 (December 24, 2006)- Which philosopher is also known as the RaMBaM?
**Urs Schreiber:**Conference: Lie Algebroids and Lie Groupoids in Differential Geometry (December 21, 2006)- Conference in Sheffield on Lie algebroids and Lie groupoids in Differential Geometry
**John Baez:**Lectures on Classical Mechanics (December 21, 2006)- Lectures notes on classical mechanics - now available in printed form!
**Urs Schreiber:**A Little Bit of Geometric Langlands: Relation to Integrable Systems (December 21, 2006)- On Hitchin integrable systems and their role in geometric Langlands duality.
**David Corfield:**Common Applications (December 21, 2006)- Why does the same piece of mathematics find many applications?
**John Baez:**Quantization and Cohomology (Week 9) (December 21, 2006)- A glimpse of geometric quantization.
**Urs Schreiber:**Excellent Math in Bonn: Opening Colloquium in January (December 20, 2006)- Opening colloquium of the new “cluster of excellence” of mathematics in Bonn.
**Urs Schreiber:**Research Blogging (December 20, 2006)- On research blogging.
**Urs Schreiber:**Universal Gerbes (December 18, 2006)- What is known about universal gerbes?
**John Baez:**This Week’s Finds in Mathematical Physics (Week 242) (December 18, 2006)- Photos of Saturn, its ring and moons. Unmanned NASA missions versus sending canned primates to Mars. Jeffrey Mortons’ work on topological quantum field theory.
**Urs Schreiber:**Frobenius Algebroids with Invertible Products (December 15, 2006)- Thinking of algebroids with invertible products as Frobenius algebroids.
**John Baez:**Higher Categories at the Fields Institute (December 14, 2006)- Schedule of talks at the Fields Institute workshop on “Higher Categories and Their Applications”
**David Corfield:**Back from NIPS 2006 (December 13, 2006)- Background knowledge in machine learning
**David Corfield:**Bicat is Not Triequivalent to Gray (December 13, 2006)- Steve Lack’s new paper.
**Urs Schreiber:**Communicating Thoughts on the Web (December 13, 2006)- Thinking, communicating and having math on the web.
**Urs Schreiber:**Navigating Geometric Langlands by Analogies (December 12, 2006)- Some basic analogies that help us navigate the geometric Langlands duality.
**Urs Schreiber:**Local Transition of Transport, Anafunctors and Descent of n-Functors (December 8, 2006)- Concepts and examples of what would be called transition data or descent data for n-functors.
**John Baez:**New Light Particle(s) Discovered? (December 7, 2006)- Hints of particles with masses near 7 and 19 MeV?
**John Baez:**Guiraud on Higher-Dimensional Rewrite Rules (December 6, 2006)- The three dimensions of proof.
**Urs Schreiber:**Talks at “Higher Categories and their Applications” (December 5, 2006)- Talks on 2-functorial TFT and CFT.
**Urs Schreiber:**Postdoctoral Position at the Interface of Algebra, Conformal Field Theory and String Theory (December 5, 2006)- Postdoc position in algebra, CFT and strings.
**Urs Schreiber:**What Does the Classifying Space of a 2-Category Classify? (December 4, 2006)- Notes by Baas, Boekstedt and Kro on classifying spaces for 2-vector bundles.
**David Corfield:**Klein 2-Geometry VIII (December 1, 2006)- Continuing Klein 2-geometry
**John Baez:**Classical vs Quantum Computation (Week 8) (December 1, 2006)- How Curry’s Fixed Point Theorem is related to Cantor’s diagonal argument.
**John Baez:**Quantization and Cohomology (Week 8) (December 1, 2006)- From particles to membranes, continued.
**John Baez:**Puzzle #8 (November 30, 2006)- Which mathematician was rumored in 1999 to be secretly in charge of one of the world’s largest countries?
**Urs Schreiber:**Nicolai on E10 and Supergravity (November 29, 2006)- H. Nicolai on further progress in checking the hypothesis that the dynamics of supergravity is encoded in geodesic motion on a Kac-Moody group coset.
**Urs Schreiber:**D-Branes from Tin Cans, II (November 28, 2006)- Gerbe modules from 2-sections.
**David Corfield:**NIPS 2006 (November 27, 2006)- NIPS 2006 conference, Vancouver
**John Baez:**Puzzle #7 (November 25, 2006)- Which bird can sleep with half its brain while the other half stays awake?
**Urs Schreiber:**2-Monoid of Observables on String-G (November 24, 2006)- Rep(L G) from 2-sections.
**Urs Schreiber:**The Baby Version of Freed-Hopkins-Teleman (November 23, 2006)- The Freed-Hopkins-Teleman result and its baby version for finite groups, as explained by Simon Willerton.
**Urs Schreiber:**The 1-Dimensional 3-Vector Space (November 23, 2006)- Alg(C) as the 3-category of canonical 1-dimensional 3-vector spaces.
**John Baez:**A Third Model of the String Lie 2-Algebra (November 23, 2006)- Wagemann gives a third construction of the one-parameter family of Lie 2-algebras associated to any simple Lie algebra.
**John Baez:**Classical vs Quantum Computation (Week 7) (November 21, 2006)- Building a computer using the lambda calculus: the Fixed Point Theorem.
**Urs Schreiber:**Basic Question on Homs in 2-Cat (November 21, 2006)- A basic question on internal homs in 2-categories.
**David Corfield:**Philosophy as Stance (November 21, 2006)- Philosophy and taking a stance.
**Urs Schreiber:**Derived Categories in Utah (November 20, 2006)- A workshop on derived categories and their application in string theory.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 241) (November 20, 2006)- Gravitational wave detection in Louisiana, higher gauge theory and the dodecahedron.
**Urs Schreiber:**Categorical Trace and Sections of 2-Transport (November 17, 2006)- A general concept of extended QFT and its relation to the Kapranov-Ganter 2-character.
**David Corfield:**MacIntyre on Rational Judgment (November 16, 2006)- How explicit must rational reasoning be?
**John Baez:**Classical vs Quantum Computation (Week 6) (November 15, 2006)- From lambda-terms to string diagrams. Building a computer inside a cartesian closed category with an object X with X ≅ hom(X,X).
**John Baez:**Quantization and Cohomology (Week 7) (November 15, 2006)- Generalizing classical mechanics from particles to strings and higher-dimensional membranes.
**John Baez:**Higher Gauge Theory (November 15, 2006)- Transparencies for a talk on higher gauge theory.
**John Baez:**Breen on Gerbes and 2-Gerbes (November 13, 2006)- A new expository paper.
**John Baez:**Quantization and Cohomology (Week 6) (November 13, 2006)- The canonical 1-form, extended phase space and Rovelli’s covariant formulation of classical mechanics.
**John Baez:**Quantization and Cohomology (Week 5) (November 13, 2006)- The canonical 1-form on the cotangent bundle of a manifold, and what it does for classical mechanics.
**John Baez:**Puzzle #6 (November 13, 2006)- Which famous buildings were named after a form of food - or was it the other way around?
**John Baez:**Tales of the Dodecahedron (November 12, 2006)- Slides from a talk on the history of the dodecahedron, from Pythagoras through Plato to Poincaré.
**David Corfield:**The Tasks of Philosophy (November 10, 2006)- What should philosophy study?
**David Corfield:**Mathematics Under the Microscope (November 9, 2006)- Book about mathematics
**Urs Schreiber:**Flat Sections and Twisted Groupoid Reps (November 8, 2006)- A comment on Willerton’s explanation of twisted groupoid reps in terms of flat sections of n-bundles.
**John Baez:**Quantization and Cohomology (Week 4) (November 8, 2006)- Hamiltonian dynamics and symplectic geometry.
**John Baez:**Quantization and Cohomology (Week 3) (November 8, 2006)- From Lagrangian to Hamiltonian dynamics.
**Urs Schreiber:**Chern-Simons Lie-3-Algebra Inside Derivations of String Lie-2-Algebra (November 7, 2006)- The Chern-Simons Lie 3-algebra sits inside that of inner derivations of the string Lie 2-algebra.
**Urs Schreiber:**Dijkgraaf-Witten Theory and its Structure 3-Group (November 6, 2006)- The idea of Dijkgraaf-Witten theory and its reformulation in terms of parallel volume transport with respect to a structure 3-group.
**David Corfield:**Infinite-Dimensional Exponential Families (November 6, 2006)- Information geometry for kernel methods in machine learning.
**John Baez:**Puzzle #5 (November 5, 2006)- When was the Roman empire sold, and who bought it?
**Urs Schreiber:**A 3-Category of Twisted Bimodules (November 3, 2006)- A 3-category of twisted bimodules.
**John Baez:**Classical vs Quantum Computation (Week 5) (November 2, 2006)- The naturality of currying - and a new "bubble" notation for currying and uncurrying.
**John Baez:**A Categorical Manifesto (November 2, 2006)- Why are categories important in computer science?
**David Corfield:**Academic Commons (November 2, 2006)- Enclosure of Academic Commons
**David Corfield:**Klein 2-Geometry VII (November 1, 2006)- Continuing to categorify the Erlanger Program
**Urs Schreiber:**Bakovic on 2-Groupoid 2-Torsors (October 31, 2006)- Igor Bakovic defines and classifies categorified groupoid bundles.
**Urs Schreiber:**Markl on Natural Differential Operators (October 31, 2006)- M. Markl explains how to think about “naturality” of differential operators.
**David Corfield:**Foundations (October 30, 2006)- What are foundations?
**John Baez:**Puzzle #4 (October 30, 2006)- Such a brivla, built from the rafsi for the component gismu and cmavo, is called a lujvo.
**Urs Schreiber:**WZW as Transition 1-Gerbe of Chern-Simons 2-Gerbe (October 29, 2006)- How the WZW 1-gerbe arises as the transition 1-gerbe of the Chern-Simons 2-gerbe.
**Urs Schreiber:**Hopkins Lecture on TFT: Chern-Simons (October 27, 2006)- Basic and advanced concepts in Chern-Simons theory.
**John Baez:**Classical vs Quantum Computation (Week 4) (October 26, 2006)- Currying and uncurrying, evaluation and coevaluation: basic aspects of the "quantum lambda-calculus".
**John Baez:**Lauda on Topological Field Theory and Tangle Homology (October 25, 2006)- Aaron Lauda’s thesis on topological field theory and tangle homology.
**Urs Schreiber:**Hopkins Lecture on TFT: Infinity-Category Definition (October 25, 2006)- A suggestion for a definition of (extended) TFT in terms of morphisms of omega-categories.
**Urs Schreiber:**Hopkins Lecture on TFT: Introduction and Outlook (October 25, 2006)- Introductory lecture by M. Hopkins on topological field theory.
**John Baez:**Puzzle #3 (October 25, 2006)- A special law to prevent votes from being counted.
**David Corfield:**Knowledge of the Reasoned Fact (October 23, 2006)- In a comment I raised the question of what to make of our expectation that behind different manifestations of an entity there is one base account, of which these manifestations are consequences. If I point out to you three…
**John Baez:**This Week’s Finds in Mathematical Physics (Week 240) (October 23, 2006)- Read about Dolan and Trimble’s work on cartesian closed categories and
*holodeck games!* **John Baez:**Classical vs Quantum Computation (Week 3) (October 20, 2006)- Guest lecture by James Dolan: Holodeck strategies and cartesian closed categories.
**Urs Schreiber:**Algebras as 2-Categories and its Effect on Algebraic Geometry (October 19, 2006)- A question by Bruce Bartlett about categories of algebras, algebras as categories and the possible implications for non-commutative algebraic geometry.
**John Baez:**A Course on Classical Mechanics (October 19, 2006)- A course on classical mechanics from a mathematical viewpoint.
**John Baez:**Australian Category Theory (October 18, 2006)- What’s happening at the Australian Category Seminar.
**Urs Schreiber:**Topology in Trondheim and Kro, Baas & Bökstedt on 2-Vector Bundles (October 18, 2006)- Baas, Dundas and Kro with new insights into 2-vector bundles.
**Urs Schreiber:**D-Branes from Tin Cans: Arrow Theory of Disks (October 18, 2006)- On disk holonomy and boundary conditions.
**John Baez:**Classical vs Quantum Computation (Week 2) (October 17, 2006)- An intro to closed monoidal categories.
**Guest:**Crossed Products and Not All That (October 16, 2006)- A. Rivero on crossed products of C*-algebras, with an eye on Connes’ spectral description of the standard model.
**Urs Schreiber:**Adjunctions and String Composition (October 16, 2006)- A remark on describing string composition and string field products in terms of adjunctions in categories of paths.
**David Corfield:**The Encyclopaedists’ Dream (October 15, 2006)- What is epistemology?
**Urs Schreiber:**Bundle Gerbes: Connections and Surface Transport (October 13, 2006)- Definition of connection on a bundle gerbe and of the surface transport computed from it.
**Urs Schreiber:**Bundle Gerbes: General Idea and Definition (October 13, 2006)- The general idea of bundle gerbes and the basic definition.
**Urs Schreiber:**What is the Categorified Gelfand-Naimark Theorem? (October 11, 2006)- A lightning review of the categorified Gelfand-Naimark theorem.
**David Corfield:**Wittgenstein and Thurston on Understanding (October 11, 2006)- What Wittgenstein might have said about the famous Jaffe-Quinn debate - and what Thurston
*did*say. **David Corfield:**Euler Characteristic of a Category (October 11, 2006)- Baez and Dolan defined the cardinality of a groupoid. Now Tom Leinster has defined a more general concept: the Euler characteristic of a category!
**John Baez:**Categorified Gelfand-Naimark Theorem and Vector Bundles with Connection (October 11, 2006)- A question from Bruce Bartlett
**John Baez:**Quantization and Cohomology (Week 2) (October 11, 2006)- The Lagrangian approach to classical mechanics
**Urs Schreiber:**Talks by Hopkins and Willerton (October 10, 2006)- Talk by Hopkins of TFT and by Willerton on Dijkgraaf-Witten.
**Urs Schreiber:**Kosmann-Schwarzbach & Weinstein on Lie Algebroid Classes (October 10, 2006)- Kosmann-Schwarzbach on classes of Lie algebroids generalizing the character of a Lie algebra.
**Urs Schreiber:**On n-Transport: Descent of the Universal Transition (October 9, 2006)- How the universal transition mediates descent from a surjection down to the original space.
**David Corfield:**Philosophy of Physics (October 9, 2006)- Publication of ‘Philosophy of Physics’
**John Baez:**Zooming Out in Time (October 8, 2006)- To understand what a few degrees of global warming can do, we must study the Earth’s climate over the last 65 million years.
**John Baez:**Classical vs Quantum Computation (Week 1) (October 6, 2006)- Types and operations; categories as theories; monoidal categories versus categories with finite products.
**Urs Schreiber:**On n-Transport: Universal Transition (October 6, 2006)- Paths in categories from universal transitions.
**David Corfield:**So Irigaray was Right? (October 6, 2006)- Navier-Stokes solution
**John Baez:**Quantization and Cohomology (Week 1) (October 5, 2006)- How the dynamics of p-branes resembles the statics of (p+1)-branes.
**John Baez:***New Scientist*Reacts! (October 5, 2006)*New Scientist*now has a blog entry defending their article on Shawyer’s electromagnetic drive.**John Baez:**Trimble’s Definition of Tetracategory (October 5, 2006)- Todd Trimble has redrawn his legendary lost definition of tetracategories.
**John Baez:**λ-Calculus in Paris (October 4, 2006)- Typed Lambda Calculi and Applications - a conference in Paris, June 26-28, 2007
**David Corfield:**Emulating Aristotle (October 4, 2006)- How to emulate Aristotle
**Guest:**Wang at AustMS 2006 (October 4, 2006)- The following “guest post” is by David Roberts: At the Australian Mathematical Society’s annual meeting in Sydney, Brian Wang gave the talk Gerbes, D-branes and loop group representations. This is of particular interest to at least one blogger here, so…
**John Baez:**Classical vs Quantum Computation (Week 1) (October 3, 2006)- Computation, the lambda calculus and cartesian closed categories: an overview.
**John Baez:**Voevodsky on the Homotopy λ-Calculus (October 2, 2006)- Voevodsky has released a “very short note on the homotopy λ-calculus”.
**John Baez:**Topos Theory at Chicago (October 2, 2006)- Lectures on topos theory at Chicago
**David Corfield:**The Consolation of n-Categories (October 1, 2006)- What has mathematics meant to people through the ages?
**David Corfield:**Klein 2-Geometry VI (October 1, 2006)- Resuming the categorification of Kleinian Geometry
**John Baez:**Puzzle #2 (October 1, 2006)- A spooky link between Tycho Brahe and Shakespeare.
**John Baez:**Dimensional Analysis and Coordinate Systems (September 30, 2006)- More gnarly issues: dimensional analysis and coordinate systems.
**Urs Schreiber:**Puzzle Pieces Falling Into Place (September 28, 2006)- On the 3-group which should be underlying Chern-Simons theory.
**Urs Schreiber:**2-Groups and Algebras (September 28, 2006)- Bundles of algebras from principal 2-group transitions.
**David Corfield:**Toleration (September 27, 2006)- Toleration and the ends of conversation
**Urs Schreiber:**Bulk Fields and Induced Bimodules (September 27, 2006)- Bulk field insertions in 2D CFT in terms of 2-transport: endomorphisms of 2-monoids.
**Urs Schreiber:**Fahrenberg and Raussen on Continuous Paths (September 26, 2006)- A theory of continuous n-paths modulo continuous reparameterization.
**David Corfield:**Our Raison D’être (September 26, 2006)- Marni Sheppeard reports from the AustMS2006 conference, which, as anyone who knows about Australian mathematics might expect, is holding a category theory session. Dominic Verity is giving one of the talks, in which he considers the raison d’être for higher…
**David Corfield:**Mathematical Kinds (September 23, 2006)- Do mathematical entities belong to kinds?
**John Baez:**Dimensional Analysis (September 21, 2006)- Let’s amass a collection of wisdom on gnarly issues in physics - starting with dimensional analysis.
**David Corfield:**Categorification in Uppsala (September 21, 2006)- Earlier this month the Mathematics Institute at Uppsala University hosted a conference called Categorification in Algebra and Topology, clearly a theme close to our collective heart. As yet there are only a handful of participants’ notes available (Scott Morrison’s are…
**David Corfield:**The Why and Wherefore of History (September 21, 2006)- Here are some notes for my talk at the Berlin workshop. Fortunately I was upgraded to a 45-minute talk. Even so, I didn’t manage to reach the last part where I discuss David Carr’s ideas. I would be interested in…
**Urs Schreiber:**Differential n-Geometry (September 20, 2006)- A quest for arrow-theoretic differential geometry.
**John Baez:**A Plea to Save*New Scientist*(September 19, 2006)- The SF writer Greg Egan has issued a plea to save the magazine New Scientist.
**David Corfield:**Searching for a New Epistemology in Berlin (September 18, 2006)- Reporting from a workshop in Berlin.
**Urs Schreiber:**Quantum n-Transport (September 14, 2006)- An attempt to understand the path integral for an n-dimensional field theory as a coproduct operation over transport n-functors.
**John Baez:**Higher Categories and their Applications (September 13, 2006)- In January 2007, the Fields Institute is having a workshop on Higher Categories and their Applications.
**John Baez:**Groupoids and Stacks in Physics and Geometry (September 13, 2006)- In 2007, the Institut Henri Poincaré will be running a program on Groupoids and Stacks in Physics and Geometry.
**Urs Schreiber:**Freed on Higher Structures in QFT, I (September 12, 2006)- Freed’s old observation that n-dimensional QFT assigns (n-d)-Hilbert spaces to d-dimensional volumes.
**Urs Schreiber:**Wirth and Stasheff on Homotopy Transition Cocycles (September 11, 2006)- Stasheff recalls an old result by Wirth on passing between fibrations and their homotopy transition cocycles.
**David Corfield:**Mathematics and Virtual Reality (September 9, 2006)- Is Mathematics a Massively-Multiplayer Online Role-Playing Game?
**Urs Schreiber:**Kock on 1-Transport (September 8, 2006)- A new preprint by Anders Kock on the synthetic formulation of the notion of parallel transport.
**Urs Schreiber:**Connes on Spectral Geometry of the Standard Model, IV (September 8, 2006)- The details of the spectral triple which describes the standard model.
**John Baez:**This Week’s Finds in Mathematical Physics (Week 239) (September 8, 2006)- Read about the open access movement, Freeman Dyson’s 1951 lecture notes, the origins of mathematics in little clay figures called “tokens”, and Koszul duality for L
^{∞}-algebras! **David Corfield:**Category Theory and Philosophy (September 8, 2006)- The place of category theory in philosophy
**Urs Schreiber:**Connes on Spectral Geometry of the Standard Model, III (September 7, 2006)- First technical details for spectral action theory.
**Urs Schreiber:**On n-Transport: 2-Vector Transport and Line Bundle Gerbes (September 7, 2006)- Associated 2-transport, 2-representations and bundle gerbes with connection.
**David Corfield:**Mathematical Circuit Components (September 7, 2006)- What mathematics and radio circuits might have in common.
**Urs Schreiber:**Connes on Spectral Geometry of the Standard Model, II (September 6, 2006)- On the nature and implication of Connes’ identification of the spectral triple of the standard model coupled to gravity.
**Urs Schreiber:**Connes on Spectral Geometry of the Standard Model, I (September 6, 2006)- Connes is getting closer to the spectral triple encoding the standard model coupled to gravity. Part I: some background material.
**Urs Schreiber:**n-Transport and Higher Schreier Theory (September 5, 2006)- Understanding n-transport in terms of Schreier theory for groupoids.
**David Corfield:**The History of n-Categories (September 4, 2006)- How to write the history of mathematics.
**David Corfield:**Klein 2-Geometry V (September 4, 2006)- We resume the attempt to categorify Kleinian geometry
**John Baez:**Doctrines (September 3, 2006)- In 1969 Lawvere invented “doctrines”. A doctrine is a roughly a kind of category, thought of as a kind of logical theory.
**John Baez:**TeXnical Issues (September 2, 2006)- Having trouble reading the equations on this blog, or posting comments? Try this.
**John Baez:**Ringoids (September 2, 2006)- A group with many objects is a groupoid. A ring with many objects is a ringoid!
**John Baez:**Letter from Grothendieck (August 31, 2006)- Newly released letter from Alexander Grothendieck.
**Urs Schreiber:**10D SuGra 2-Connection (August 28, 2006)- On the Lie 2-algebra governing 10-dimensional supergravity.
**John Baez:**Quantum Computation and Symmetric Monoidal Categories (August 28, 2006)- Ordinary computer scientists like cartesian closed categories. Quantum computer scientists like symmetric monoidal categories with duals.
**John Baez:**Categorifying CCCs: Computation as a Process (August 28, 2006)- “Cartesian closed n-categories” may help us see computation as a process.
**John Baez:**CCCs and the λ-calculus (August 28, 2006)- Let’s discuss cartesian closed categories and the λ-calculus!
**Urs Schreiber:**Picturing Morphisms of 3-Functors (August 25, 2006)- Diagrams governing morphisms of 3-functors.
**John Baez:**Categories and Computation (August 24, 2006)- What’s the relation between the lambda-calculus, computation and CCCs?
**John Baez:**Categorifying the Dijkgraaf-Witten model (August 24, 2006)- Martins and Porter categorify the Dijkgraaf-Witten model.
**John Baez:**Puzzle #1 (August 23, 2006)- What was Uncle Sam’s last name?
**John Baez:**Lectures on n-Categories and Cohomology (August 22, 2006)- Lectures on n-Categories and Cohomology, by John Baez and Michael Shulman
**Urs Schreiber:**n-Curvature (August 21, 2006)- Definition of n-curvature and the nature of “fake curvature”.
**Urs Schreiber:**On n-Transport, Part II (August 21, 2006)- Smooth transport, differentials of transport, and nonabelian differential cocycles.
**Urs Schreiber:**On n-Transport, Part I (August 18, 2006)- The concept of n-transport and its local trivialization and transition.
**Urs Schreiber:**SuGra 3-Connection Reloaded (August 17, 2006)- Discussion of n-category description of supergravity continued.
**John Baez:**Dark Matter in the Bullet Cluster (August 17, 2006)- Hard evidence for dark matter in the Bullet Cluster
**Distler:**Inaugural Post (August 16, 2006)- Brand new blog.

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