Spans of cospans

TL;DR

This paper introduces a bicategory structure based on spans of cospans in a topos, with applications to graph rewriting, by defining compositions and conditions for their interchange law.

Contribution

It formalizes the notion of spans of cospans, defines their compositions, and constructs a bicategory applicable to graph rewriting in a topos.

Findings

Bicategory constructed from C-objects and cospans

Interchange law holds under monic span legs in a topos

Application demonstrated in graph rewriting contexts

Abstract

We discuss the notion of a span of cospans and define, for them, horizonal and vertical composition. These compositions satisfy the interchange law if working in a topos $\mathbf{C}$ and if the span legs are monic. A bicategory is then constructed from $\mathbf{C}$-objects, $\mathbf{C}$-cospans, and doubly monic spans of $\mathbf{C}$-cospans. The primary motivation for this construction is an application to graph rewriting.