Spans of cospans

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.