For a topos T, there is a bicategory MonicSp(Csp(T)) whose objects are those of T, morphisms are cospans in T, and 2-morphisms are isomorphism classes of monic spans of cospans in T. Using a result of Shulman, we prove that 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 MonicSp(Csp(Graph)).
Keywords: bicategory, graph rewrite, network, span, symmetric monoidal, topos
2010 MSC: 16B50, 18D05 and 18D10
Theory and Applications of Categories, Vol. 33, 2018, No. 1, pp 1-22.