Related papers: Basic Category Theory
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.…
Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations…
This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to…
We study (vertically) normal lax double functors valued in the weak double category $\mathbb{C}\mathrm{at}$ of small categories, functors, profunctors and natural transformations, which we refer to as lax double presheaves. We show that for…
There is some consensus among orthodox category theorists that the concept of adjoint functors is the most important concept contributed to mathematics by category theory. We give a heterodox treatment of adjoints using heteromorphisms…
This text is devoted to the theory of varieties, which provides an important tool, based in universal algebra, for the classification of regular languages. In the introductory section, we present a number of examples that illustrate and…
We introduce the notion of a diagram category and discuss its application to the invariant theory of classical groups and super groups, with some indications concerning extensions to quantum groups and quantum super groups. Tensor functors…
This paper provides a short introduction to the notion of regular category and its use in categorical algebra. We first prove some of its basic properties, and consider some fundamental algebraic examples. We then analyse the algebraic…
These are expanded lecture notes from lectures given at the Workshop on higher structures at MATRIX Melbourne. These notes give an introduction to Feynman categories and their applications. Feynman categories give a universal categorical…
We give a short introduction to category theory aimed at philosophers. We emphasize methodological issues and philosophical ramifications.
We give an elementary description of $2$-categories $\mathbf{Cat}\left(\mathcal{E}\right)$ of internal categories, functors and natural transformations, where $\mathcal{E}$ is a category modelling Lawvere's elementary theory of the category…
Lenses have a rich history and have recently received a great deal of attention from applied category theorists. We generalize the notion of lens by defining a category $\mathsf{Lens}_F$ for any category $\mathcal{C}$ and functor $F\colon…
The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of $\mathcal{F}$-quotients in a…
In this book there will be found an introduction to transcendental number theory, starting at the beginning and ending at the frontiers. The emphasis is on the conceptual aspects of the subject, thus the effective theory has been more or…
A diverse collection of fusion categories may be realized by the representation theory of quantum groups. There is substantial literature where one will find detailed constructions of quantum groups, and proofs of the…
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…
We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of…
This introduction to higher category theory is intended to a give the reader an intuition for what $(\infty,1)$-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from…
Based on rectangle theory of formal concept and set covering theory, the concept reduction preserving binary relations is investigated in this paper. It is known that there are three types of formal concepts: core concepts, relative…
Motivated by team semantics and existential second-order logic, we develop a model-theoretic framework for studying second-order objects such as sets and relations. We introduce a notion of abstract elementary team categories that…