Double Categories in Mathematical Physics

TL;DR

This paper explores double categories, a generalization of bicategories, demonstrating their presence in topological and quantum field theories and dynamical systems, thus enriching the categorical framework in mathematical physics.

Contribution

It introduces the concept of double categories as an extension of bicategories and shows their applications in various areas of mathematical physics.

Findings

Double categories are present in topological quantum field theories.

They also appear in ordinary quantum field theories.

Double categories are applicable to dynamical systems with inputs and outputs.

Abstract

Expansion of the categorical point of view on many areas of the mathematics and mathematical physics will cause to deeper understanding of genuine features of these problems. New applications of categorical methods are connected with new additional structures on categories. One of such structures, the double category, is considered in this article. The double category structure is defined as generalization of the bicategory structure. It is shown that double categories exist in the topological and ordinary quantum field theories, and for dynamical systems with inputs and outputs. Morphisms of all these double categories are not maps of sets.