Kyoto Category Theory Meeting

A strong epimorphism is a morphism having the left lifting property with respect to every monomorphism. By the small object argument, every strong epimorphism can be decomposed into a transfinite sequence of regular epimorphisms in a suitable category.

For example, in a category of small categories, every strong epimorphism can be decomposed into two regular epimorphisms. In this talk, I will present a syntactic way to calculate the supremum length of such decompositions.

In logic, regular theories are those built using equations, relation symbols, conjunctions, and existential quantification. The categories of models of such theories have been widely studied and characterised in purely category theoretical terms through (among others) the notion of injectivity class, that I will recall during the talk.

When moving to enriched categories, a generalized notion "injectivity class" has been introduced by Lack and Rosicky, but no enriched version of regular logic had been considered in the literature before. The aim of this talk, which is based on joint work with Rosicky, is to fill this gap by introducing a notion of "enriched regular logic" that interacts well with that of enriched injectivity. I will provide several examples on different bases of enrichment, and also explain how this is related to the internal logic of a topos.

Equipments, a special kind of double categories, have shown to be a powerful environment to express formal category theory. We build a model structure on the category of double categories and double functors whose fibrant objects are the equipments, and combine this together with Makkai’s early approach to equivalence invariant statements in higher category theory via FOLDS (First Order Logic with Dependent Sorts) and Henry’s recent connection between model structures and formal languages, to show a result on the equivalence invariance of formal category theory.

The study of categorical logic has developed based on doctrines, fibrations, and occasionally bicategories as semantic environments for logical systems. We propose an alternative approach using virtual double categories for this purpose. In this talk, we will contrast virtual double categories with other categorical structures intended for categorical logic. The main result is the construction of a 2-functor from the 2-category of cartesian fibrations to the 2-category of cartesian fibrational virtual double categories and a characterization of elementary existential fibrations, known as a semantic counterpart of regular logic, as fibrations that induce a cartesian equipment by this construction. 

A morphism p:X→Y in a category C with pullbacks is called exponentiable if the functor p*:C/Y→C/X, defined by pulling back along p, has a right adjoint. Giraud and Conduché characterized the exponentiable functors (i.e., the exponentiable morphisms in C=Cat) by explicit conditions. In this talk, I will present a proof of the equivalence of the exponentiability and the Giraud–Conduché condition for a functor, using nerves of categories. (Based on discussions with Steve Lack.)

We propose simple structure-axiom style definitions of opetopes and

opetopic sets. We show that our definition is equivalent to an

existing one in the literature.

There is an algebraic definition of weak higher categories due to Batanin and Leinster, which feels much closer to the strict higher categories compared to the more popular, non-algebraic models (such as iterated Segal spaces or complicial sets, which you don't have to know about to understand this talk). In a joint project with Soichiro Fujii and Keisuke Hoshino, we are attempting to understand the homotopy theory of these objects by constructing a (left semi-)model structure. In this talk, I will summarise what we have achieved so far and what remains to be done.