Double Bicategories and Double Cospans

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 in terms of bicategories internal to 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 describe examples in the form of double bicategories of “double cospans” (or “double spans”) in any category with pushouts (pullbacks, respectively). We also give a construction from this which involves taking isomorphism classes of objects, and gives a Verity double bicategory of double cospans. Finally, we describe how to use a minor variation on this to describe cobordism of manifolds with boundary.