Related papers: When is Cat(Q) cartesian closed?

A generalization of the notion of an $\infty$-category is presented, allowing for ($\infty$-)cat(egorie)s that may have non-invertible higher morphisms.

Quantum categories were introduced in [4] as generalizations of both bi(co)algebroids and small categories. We clarify details of that work. In particular, we show explicitly how the monadic definition of a quantum category unpacks to a set…

We study the monoidal closed category of symmetric multicategories, especially in relation with its cartesian structure and with sequential multicategories (whose arrows are sequences of concurrent arrows in a given category). Then we…

We prove that every finitary polynomial endofunctor of a category $C$ has a final coalgebra if $C$ is locally Cartesian closed, has finite disjoint coproducts and a natural number object. More generally, we prove that the category of…

This note explains how dependent sums and products are interpreted by adjoints of the base change functor in a locally cartesian closed category. An effort is made to unpack all the definitions so as to make the concepts more transparent to…

For a differential graded k-quiver Q we define the free A-infinity-category FQ generated by Q. The main result is that for an arbitrary A-infinity-category A the restriction A-infinity-functor A_\infty(FQ,A) -> A_1(Q,A) is an equivalence,…

We describe a sufficient condition for the process of left Kan extension to be a conservative functor. This is useful in the study of graphic Fourier transforms and quantum categories and groupoids.

In 1990, Johnstone gave a syntactic characterisation of the equational theories whose associated varieties are cartesian closed. Among such theories are all unary theories -- whose models are sets equipped with an action by a monoid M --…

As a practical foundation for a homotopy theory of abstract spacetime, we extend a category of certain compact partially ordered spaces to a convenient category of locally preordered spaces. In particular, we show that our new category is…

Categorical structures and their pseudomaps rarely form locally presentable 2-categories in the sense of Cat-enriched category theory. However, we show that if the categorical structure in question is sufficiently weak (such as the…

In this paper we present background results in enriched category theory and enriched model category theory necessary for developing model categories of enriched functors suitable for doing functor calculus.

Enrichment and internal categories are two different way to generalize the notion of category. As such, enriching double categories (which are categories internal to Cat) is not a clear concepts. One can look at the internal categories of…

In our previous papers we introduced categorical invariants, which are, roughly speaking, sets of triangulated subcategories in a given triangulated category and their quotients. Here is extended the list of examples, where these sets are…

We consider categorical logic on the category of Hilbert spaces. More generally, in fact, any pre-Hilbert category suffices. We characterise closed subobjects, and prove that they form orthomodular lattices. This shows that quantum logic is…

For a small involutive quantaloid $\mathcal{Q}$, it is shown that the category of separated complete $\mathcal{Q}$-categories and left adjoint $\mathcal{Q}$-functors is strictly monadic over the category of symmetric…

We describe a finite quantum groupoid associated to any finite Vect_k enriched herd.

We classify the "quotients" of a tannakian category in which the objects of a tannakian subcategory become trivial, and we examine the properties of such quotient categories.

We demonstrate that, in certain cases, quantization and the classical limit provide functors that are "almost inverse" to each other. These functors map between categories of algebraic structures for classical and quantum physics,…

We propose a definition of higher inductive types in $(\infty,1)$-categories with finite limits. We show that the $(\infty,1)$-category of $(\infty,1)$-categories with higher inductive types is finitarily presentable. In particular, the…

In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory.…