## Archival Version

**These pages are not updated anymore.
They reflect the state of
.
For the current production of this journal, please refer to
http://www.springer.com/mathematics/geometry/journal/40062.
**

Double Bicategories and Double Cospans
#
Double Bicategories and Double Cospans

##
Jeffrey C. Morton

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