Related papers: Cartesian bicategories with choice
We define a bicategory with \'etale, locally compact groupoids as objects and suitable correspondences, that is, spaces with two commuting actions as arrows; the 2-arrows are injective, equivariant continuous maps. We prove that the usual…
We construct an internal language for cartesian closed bicategories. Precisely, we introduce a type theory modelling the structure of a cartesian closed bicategory and show that its syntactic model satisfies an appropriate universal…
2-Theories are a canonical way of describing categories with extra structure. 2-theory-morphisms are used when discussing how one structure can be replaced with another structure. This is central to categorical coherence theory. We place a…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
We define bicategories internal to 2-categories. When the ambient 2-category is symmetric monoidal categories, this provides a convenient framework for encoding the structures of a symmetric monoidal 3-category. This framework is well…
In a bicategory of spans (an example of a 'generic bicategory') the factorization of a span (s,t) as the span (s,1) followed by (1,t) satisfies a simple universal property with respect to all factorizations in terms of the generic…
Binary multirelations generalise binary relations by associating elements of a set to its subsets. We study the structure and algebra of multirelations under the operations of union, intersection, sequential and parallel composition, as…
Many types of categorical structure obey the following principle: the natural notion of equivalence is generated, as an equivalence relation, by identifying $A$ with $B$ when there exists a strictly structure-preserving map $A \to B$ that…
A concise guide to very basic bicategory theory, from the definition of a bicategory to the coherence theorem.
We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over…
Recently, there has been growing interest in bicategorical models of programming languages, which are "proof-relevant" in the sense that they keep distinct account of execution traces leading to the same observable outcomes, while assigning…
We show how the notion of intercategory encompasses a wide variety of three-dimensional structures from the literature, notably duoidal categories, monoidal double categories, cubical bicategories, double bicategories and Gray categories.…
In this thesis, we introduce Cartesian double categories, motivated by the work of Carboni, Kelly, Walters, and Wood on Cartesian bicategories. Moving from bicategories to the slightly more generalized notion of double categories allows us…
We interpret several constructions with C*-algebras as colimits in the bicategory of correspondences. This includes crossed products for actions of groups and crossed modules, Cuntz-Pimsner algebras of proper product systems, direct sums…
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…
Various concerns suggest looking for internal co-categories in categories with strong logical structure. It turns out that in any coherent category, all co-categories are co-equivalence relations.
We investigate several categories related to transition structures, using a mixture of algebraic and topological methods. We show how two such categories are connected by a contravariant adjunction. This is the most detailed of a family of…
We give a general notion of combinatory completeness with respect to a faithful cartesian club and use it systematically to obtain characterisations of a number of different kinds of applicative system. Each faithful cartesian club…
As an example of the categorical apparatus of pseudo algebras over 2-theories, we show that pseudo algebras over the 2-theory of categories can be viewed as pseudo double categories with folding or as appropriate 2-functors into…
In this thesis I lift the Curry--Howard--Lambek correspondence between the simply-typed lambda calculus and cartesian closed categories to the bicategorical setting, then use the resulting type theory to prove a coherence result for…