Spans of cospans in a topos

Abstract

For a topos $\mathbf{T}$, there is a bicategory $\mathbf{MonicSp(Csp(T))}$ whose objects are those of $\mathbf{T}$, morphisms are cospans in $\mathbf{T}$, and 2-morphisms are isomorphism classes of monic spans of cospans in $\mathbf{T}$. Using a result of Shulman, we prove that $\mathbf{MonicSp(Csp(T))}$ is symmetric monoidal, and moreover, that it is compact closed in the sense of Stay. We provide an application which illustrates how to encode double pushout rewrite rules as $2$-morphisms inside a compact closed sub-bicategory of $\mathbf{MonicSp(Csp(Graph))}$.