Interest in weak cubical $n$-categories arises invarious contexts, in particular in topological field theories. Inthis paper, we describe a concept of \textit{\db} in terms ofbicategories internal to $\catname{Bicat}$. We show that in a specialcase one can reduce this to what we call a \textit{\vdb}, afterDomenic Verity. This is a weakened version of a double category, inthe sense that composition in both horizontal and vertical directionssatisfy associativity and unit laws only up to (coherent) isomorphisms. We describe examples in the form of double bicategoriesof "double cospans" (or "double spans") in any category withpushouts (pullbacks, respectively). We also give a construction fromthis which involves taking isomorphism classes of objects, and gives a{\vdb} of double cospans. Finally, we describe how to use a minor variation on this to describe cobordism of manifolds with boundary.
Journal of Homotopy and Related Structures, Vol. 4(2009), No. 1, pp. 389-428