Posts by Mike Shulman

Superextensive sites (May 21, 2012)
“Superextensive sites” allow us to replace covering families by singleton covers… sometimes.
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.”
The mysterious nature of right properness (May 7, 2012)
What does right properness of a model category mean?
PSSL 93 trip report (Apr 25, 2012)
A report on several talks from the PSSL 93 in Cambridge.
Exact completions and small sheaves (Mar 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.
The Multiplicativity of Fixed-Point Invariants (Mar 6, 2012)
A bicategorical point of view on fixed-point invariants implies a general multiplicativity theorem.
Propositions as Some Types and Algebraic Nonalgebraicity (Jan 12, 2012)
Different ways to relate proofs with constructions in foundational theories, and why homotopy type theory makes the choice it does.
Cellularity in Algebraic Model Structures (Dec 9, 2011)
Constructing algebraic versions of model structures, Quillen adjunctions, and monoidal model structures using cellularity of the generators.
Reflective Subfibrations, Factorization Systems, and Stable Units (Dec 6, 2011)
When can a reflective subcategory be extended to a reflective subfibration or a stable factorization system?
Internalizing the External, or The Joys of Codiscreteness (Nov 29, 2011)
An internal axiomatization of factorization systems, subtoposes, local toposes, and cohesive toposes in homotopy type theory, using “higher modality” and codiscreteness.
Discreteness, Concreteness, Fibrations, and Scones (Nov 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.
Productive Homotopy Pullbacks (Nov 7, 2011)
Certain squares built out of cartesian products are always homotopy pullbacks, in any derivator.
Traces in Indexed Monoidal Categories (Nov 4, 2011)
Indexed monoidal categories and their string diagrams provide a nice framework for generalized fixed point theory.
Fixed Point Indices for Groupoids (Aug 20, 2011)
Can you define the fixed-point index for finitely generated free groupoids in a purely categorical way?
Coinductive Definitions (Jul 30, 2011)
Coinductive definitions are just as basic as inductive ones, and are an important tool for working with ω-categories.
Flat Functors and Morphisms of Sites (Jun 19, 2011)
Two classical notions of flat functor have a common generalization, leading to a better category of sites.
Homotopy Type Theory, VI (Apr 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.
Enhanced 2-categories and Limits for Lax Morphisms (Apr 13, 2011)
What 2-categorical limits are there in the 2-category of algebras and lax morphisms for a 2-monad?
Homotopy Type Theory, V (Apr 12, 2011)
What’s still missing to make homotopy type theory into a complete internal language for (∞,1)-topoi?
Homotopy Type Theory, IV (Apr 5, 2011)
Voevodsky’s “univalence axiom” for universes in homotopy type theory, and the equivalent principle of “equivalence induction.”
Homotopy Type Theory, III (Mar 24, 2011)
Some more discussion of the notions of “equivalence” in homotopy type theory.
Homotopy Type Theory, II (Mar 18, 2011)
Continuing from last time, we define sets and logic in terms of homotopy type theory.
Homotopy Type Theory, I (Mar 11, 2011)
An overview of the correspondence between homotopical structures and intensional type theory.
Concrete ∞-Categories (Feb 16, 2011)
Is there a notion of “concrete ∞-category”?
Punctual Local Connectedness (Feb 11, 2011)
A new paper by Peter Johnstone shows that Lawvere’s “pieces have points” axiom has some surprisingly strong consequences.
Topos Theory Can Make You a Predicativist (Jan 23, 2011)
Working with higher topoi, and being used to negative thinking, can make predicative mathematics seem both more appealing and more feasible.
Internal Categories, Anafunctors and Localisations (Nov 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 n-category cafe visitors (see…
Locally Constant Sheaves (Nov 25, 2010)
What is the right definition of a locally constant sheaf on a non-locally-connected space?
Universe Enlargement (Nov 23, 2010)
Ways to enlarge a category to live in a different universe.
Petit (∞,1)-Toposes (Oct 26, 2010)
(∞,1)-toposes, regarded as generalized spaces, provide a context in which to relate spaces to ∞-groupoids in a universal way.
Structure-Like Stuff (Oct 4, 2010)
The notion of ‘property-like structure’ suggests a categorified version of ‘structure-like stuff’ – are there any examples?
Grothendieck-Maltsiniotis ∞-categories (Sep 15, 2010)
Georges Maltsiniotis has posted a new definition of ∞-category, whose groupoidal version is said to be due to Grothendieck.
The Geometry of Monoidal Fibrations? (Aug 26, 2010)
Does it exist?
LaTeX Overflow (Aug 4, 2010)
A new question+answer site for TeX questions
Ternary Factorization Systems (Jul 24, 2010)
Defining ternary factorization systems in terms of binary ones.
What is a Theory? (Jul 14, 2010)
Is a syntactically presented theory as important as its walking model?
Exact Squares (Jun 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.
Tensor Categories in Fredericton (Jun 7, 2010)
Some highlights from the Tensor Categories session at the CMS 2010 summer meeting in Fredericton.
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.
Understanding the Homotopy Coherent Nerve (Apr 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.
Confessions of a Higher Category Theorist (Apr 29, 2010)
Why might we continue to search for more models for higher categories, aside from the simplicial ones that have proven so useful?
Symmetric Monoidal Bicategories and (n×k)-categories (Apr 8, 2010)
Monoidal bicategories can be constructed easily from “fibrant” monoidal double categories, and likewise for higher n-categories.
Extraordinary 2-Multicategories (Mar 30, 2010)
The still-hypothetical notion of “extraordinary 2-multicategory” generalizes a 2-category just enough to include extraordinary natural transformations.
Stack Semantics (Mar 23, 2010)
The stack semantics extends the internal logic of a topos to handle unbounded quantifiers in a category-theoretic way.
In Praise of Dependent Types (Mar 3, 2010)
Dependent types should be everyone’s friend!
Equivariant Stable Homotopy Theory (Jan 24, 2010)
Trying to understand equivariant stable homotopy theory from a higher-categorical perspective.
Generalized Multicategories (Jan 4, 2010)
A brief introduction to generalized multicategories, in honor of a new draft of a paper about them.
The Problem With Lax Functors (Dec 12, 2009)
There is no 3-category which includes 2-categories, lax functors, and any of the usual sorts of transformations.
Syntax, Semantics, and Structuralism II (Dec 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.
Size, Yoneda, and Limits of Algebras (Dec 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.
Feferman Set Theory (Nov 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.
Equipments (Nov 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.”
Combinatorial-Game Categories (Nov 16, 2009)
Joyal’s category of Conway games is the initial “combinatorial-game category.” What is the terminal one?
Generalized Operads in Classical Algebraic Topology (Oct 30, 2009)
Two obscure-seeming kinds of generalized operad help make categorical sense out of two constructions in classical algebraic topology.
Aesthetics of Commutative Diagrams (Oct 21, 2009)
How to lay out a large commutative diagram aesthetically?
Syntax, Semantics, and Structuralism, I (Oct 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.
Associativity Data in an (∞,1)-Category (Oct 8, 2009)
Guest post by Emily Riehl A popular slogan is that (∞,1)-categories (also called quasi-categories or ∞-categories) sit somewhere between categories and spaces, combining some of the features of both. The analogy with spaces is fairly clear, at least to…