A Double Bicategory of Cobordisms with Corners

Interest in cobordism categories arises in areas from topology to theoretical physics, and in particular in Topological Quantum Field Theories (TQFT’s). These categories have manifolds as objects, and cobordisms between them as morphisms, have that is, manifolds of one dimension higher whose boundary decomposes into the source and target. Since the boundary of a boundary is empty, this formulation cannot account for cobordisms between manifolds with boundary. This is needed to describe open-closed TQFT’s, and more generally, “extended TQFT’s”. We describe a framework for describing these, in the form of what we call a Verity double bicategory. This is similar to a double category, but with properties holding only up to certain 2-morphisms. We show how a broad general class of examples arise from a construction involving spans (or cospans) in suitable settings, and how this gives cobordisms between cobordisms when we start with the category of manifolds.