Double Bicategories and Double Cospans

TL;DR

This paper introduces the concept of double bicategories, a weakened form of double categories, and demonstrates their construction using double cospans in categories with pushouts, with applications to cobordisms with corners.

Contribution

It defines double bicategories and Verity double bicategories, and constructs examples using double cospans, linking them to topological field theories and cobordisms with corners.

Findings

Double bicategories generalize double categories with weaker associativity.

Construction of double bicategories using double cospans in categories with pushouts.

Application to cobordisms with corners as an example.

Abstract

Interest in weak cubical n-categories arises in various contexts, in particular in topological field theories. In this paper, we describe a concept of double bicategory, namely a strict model of the theory of bicategories in Bicat. We show that in a special case one can reduce this to what we call a Verity double bicategory, after Domenic Verity. This is a weakened version of a double category, in the sense that composition in both horizontal and vertical directions satisfy associativity and unit laws only up to (coherent) isomorphisms. We prove that there are examples in the form of double bicategories of "double cospans" (or "double spans") in any category with pushouts (pullbacks, respectively), and give a construction from this which involves taking isomorphism classes of objects, and gives a Verity double bicategory of double cospans. We describe cobordisms with corners as an example of this construction.