Double Bicategories and Double Cospans

Details Double Bicategories and Double Cospans Double Bicategories and Double Cospans

2006-11-30

arxiv.org/abs/math/0611930v3

Journal of Homotopy and Related Structures, Vol. 4(2009), No. 1, pp. 389-428

Mathematics

Category Theory

35 Pages: v3 amends presentation of main defn, develops cobordism example, adds 2 figures

Scientific paper

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, namely a strict model of the theory of bicategories in 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 prove that there are examples in the form of double bicategories of "double cospans" (or "double spans") in any category with pushouts (pullbacks, respectively), and give a construction from this which involves taking isomorphism classes of objects, and gives a Verity double bicategory of double cospans. We describe cobordisms with corners as an example of this construction.

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