Related papers: A 2-categories companion

Higher-dimensional category theory is the study of n-categories, operads, braided monoidal categories, and other such exotic structures. Although it can be treated purely as an algebraic subject, it is inherently topological in nature: the…

We propose an abstract notion of a type theory to unify the semantics of various type theories including Martin-L\"{o}f type theory, two-level type theory and cubical type theory. We establish basic results in the semantics of type theory:…

We give a short introduction to category theory aimed at philosophers. We emphasize methodological issues and philosophical ramifications.

We prove that a (lax) bilimit of a 2-functor is characterized by the existence of a limiting contraction in the 2-category of (lax) cones over the diagram. We also investigate the notion of bifinal object and prove that a (lax) bilimit is a…

Bimorphic lenses are a simplification of polymorphic lenses that (like polymorphic lenses) have a type defined by 4 parameters, but which are defined in a monomorphic type system (i.e. an ordinary category with finite products). We show…

Categories of polymorphic lenses in computer science, and of open games in compositional game theory, have a curious structure that is reminiscent of compact closed categories, but differs in some crucial ways. Specifically they have a…

Bicategories of spans are characterized as cartesian bicategories in which every comonad has an Eilenberg-Moore ob ject and every left adjoint arrow is comonadic.

These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.

I discuss the general formalism of two-dimensional topological field theories defined on open-closed oriented Riemann surfaces, starting from an extension of Segal's geometric axioms. Exploiting the topological sewing constraints allows for…

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…

This paper aims to show that a simple framework, utilizing basic formalisms from set theory and category theory, can clarify and inform our theories of the relation between mind and matter.

We develop a purely set-theoretic formalism for binary trees and binary graphs. We define a category of binary automata, and display it as a fibred category over the category of binary graphs. We also relate the notion of binary graphs to…

The study of complex systems through the lens of category theory consistently proves to be a powerful approach. We propose that cognition deserves the same category-theoretic treatment. We show that by considering a highly-compact cognitive…

We define here the category of partial differential equations. Special cases of morphisms from an object (equation) are symmetries of the equation and reductions of the equation by a symmetry groups, but there are many other morphisms. We…

We give an elementary exposition of some fundamental facts about fibered (or rather opfibered) categories, in terms of monads and 2-categories. The account avoids any mention of category-valued functors and pseudofunctors.

Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…

We explain some applications of bicategories in both classical and quantum field theory. This includes a modern perspective on some pioneering work of Max Kreuzer and Bert Schellekens on rational conformal field theory.

We argue that category theory should become a part of the daily practice of the physicist, and more specific, the quantum physicist and/or informatician. The reason for this is not that category theory is a better way of doing mathematics,…

In these lecture notes, we give a brief introduction to some elements of category theory. The choice of topics is guided by applications to functional programming. Firstly, we study initial algebras, which provide a mathematical…

We generalize proarrow equipments from strict category theory to the $\infty$-categorical setting, introducing the concept of $\infty$-equipments. These are specific double $\infty$-categories that support an internal higher category…