A group blog on math, physics and philosophy

- Announcing the 2018 Talbot Workshop: Model-Independent Theory of Infinity-Categories (Jan 25, 2018)
- Announcing the 2018 Talbot Workshop: Model-Independent Theory of Infinity-Categories
- Internal Languages of Higher Categories II (Nov 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…
- What is the Comprehension Construction? (Jul 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).
- 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.
- 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
- 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
- On Clubs and Data-Type Constructors (Apr 17, 2017)
- An introduction to clubs via Kelly’s “On Clubs and Data-Type Constructors” written for the Kan Extension Seminar
- Gluing Together Finite Shapes with Kelly (Apr 3, 2017)
- An exegesis of “Structures defined by limits in the enriched context, I” by Kelly written for the Kan extension seminar
- Enrichment and its Limits (Apr 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
- On the Operads of J. P. May (Mar 21, 2017)
- An introduction to Kelly’s paper on the operads of J. P. May
- Algebra Valued Functors in General and Tensor Products in Particular (Mar 7, 2017)
- A summary of Peter Freyd’s article of the same title written for the Kan Extension Seminar.
- Distributive Laws (Feb 18, 2017)
- An expository account of Beck’s “Distributive Laws” paper written for the Kan Extension Seminar
- The Category Theoretic Understanding of Universal Algebra (Feb 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.
- Category Theory in Context (Nov 16, 2016)
- Announcing the publication of “Category Theory in Context” by Emily Riehl in Dover’s Aurora: Modern Math Originals series
- The Kan Extension Seminar Returns (Oct 25, 2016)
- Announcing a second iteration of the Kan extension seminar, a graduate reading course in category theory.
- Mathematics Research Community in HoTT (Oct 4, 2016)
- A call for applications to an AMS Mathematics Research Community workshop in HoTT to be held in June 2017.
- Research Fellowships at Macquarie University (Dec 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…
- An Exact Square from a Reedy Category (Sep 30, 2015)
- Any Reedy category has a canonical square of inclusions of subcategories and this square is exact.
- Wrangling Generators for Subobjects (Aug 31, 2015)
- Describes a new categorical definition of the degree of generation of a subobject.
- Breakfast at the n-Category Café (Apr 30, 2015)
- Join Michael Harris in discussing homotopy type theory.
- A Call for Examples (Dec 28, 2014)
- A call for undergraduate-level examples illustrating categorical ideas
- A Categorical Understanding of the Proof of Cantor-Schröder-Bernstein? (Dec 7, 2014)
- In search of a categorical interpretation of the proof of the Cantor–Schroeder–Bernstein theorem.
- Kan Extension Seminar Talks at CT2014 (Jun 28, 2014)
- Kan extension seminar talks at CT2014.
- Enriched Indexed Categories, Again (Jun 27, 2014)
- Describes “enriched indexed categories” which generalize and unify enriched, internal, and indexed categories following a paper of Mike Shulman.
- Categorical Homotopy Theory (Jun 7, 2014)
- Describes the content of the new book “Categorical Homotopy Theory” just published by Cambridge University Press.
- Codescent Objects and Coherence (Jun 2, 2014)
- Summarizes the approach to coherence theorems expressed as a rectification for pseudoalgebras for a 2-monad using codescent objects, following Steve Lack.
- 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.
- On Two-Dimensional Monad Theory (Apr 28, 2014)
- Describes the approach to two-dimensional universal algebra taken in the paper of Blackwell, Kelly, and Power on two-dimensional monad theory.
- Elementary Observations on 2-Categorical Limits (Apr 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.
- On a Topological Topos (Apr 7, 2014)
- A survey of Peter Johnstone’s “On a Topological Topos”.
- An Exegesis of Yoneda Structures (Mar 24, 2014)
- Motivates the notion of Yoneda structure as an expression of basic notions of category theory in a natural 2-categorical language.
- Review of the Elements of 2-Categories (Mar 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
- Metric Spaces, Generalized Logic, and Closed Categories (Feb 21, 2014)
- Summarizes Lawvere’s “Metric Spaces, Generalized Logic, and Closed Categories” which explores applications of enriched category theory to metric topology.
- Categories of Continuous Functors (Feb 5, 2014)
- Summarizes the orthogonal subcategory problem and its solution, presented in Freyd and Kelly’s “Categories of continuous functors I”
- An Emerging Pattern in Algebra and Topology II (Feb 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.
- An Emerging Pattern in Algebra and Topology I (Feb 1, 2014)
- An emerging pattern in algebra and topology leads to a new notion of representation stability, inspired by homological stability.
- Formal Theory of Monads (Following Street) (Jan 27, 2014)
- Summarizes and contextualizes Street’s 1972 paper “The formal theory of monads.”
- An Elementary Theory of the Category of Sets (Jan 12, 2014)
- Summarizes and contextualizes Lawvere’s 1965 paper “An elementary theory of the category of sets.”
- Kan Extension Seminar applications (Nov 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….
- Announcing the Kan Extension Seminar (Oct 16, 2013)
- Announcement for an online graduate reading course in category theory.
- Mapping (Co)cylinder Factorizations via the Small Object Argument (Sep 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.
- A Weighted Limits Proof of Monadicity (Jul 10, 2013)
- The proof of Beck’s monadicity theorem is “all in the weights.”

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