The University of San Diego mathematics department is hiring! We have two assistant professor positions open this year; these are starting tenure-track positions. Applications must be through the USD web site; the deadline is November 15, 2021. The University…
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.
Interpreting linear logic into intuitionistic logic via a Chu construction automatically produces many concepts of traditional constructive mathematics.
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.
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).
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.
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.
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.
With a type-theoretic foundational system, we can describe constructions on objects with internalized universal properties, including the hom-functors of higher groupoids.
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.
Enriched indexed categories, indexed over a base category and enriched over an indexed monoidal category, simultaneously generalize indexed categories, internal categories, and enriched categories.
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.
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.
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.
Most kinds of exact completion, including categories of sheaves, are a special case of a reflection from certain sites into higher-ary exact categories.
An internal axiomatization of factorization systems, subtoposes, local toposes, and cohesive toposes in homotopy type theory, using “higher modality” and codiscreteness.
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.
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.
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…
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.
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.
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.
The still-hypothetical notion of “extraordinary 2-multicategory” generalizes a 2-category just enough to include extraordinary natural transformations.
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.
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.
(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.
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.”
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…