un 2 00 8 Double Categories in Mathematical Physics

Expansion of the categorical point of view on many areas of the mathematics and mathematical physics will cause to deeper understanding of genuine features of these problems. New applications of categorical methods are connected with new additional structures on categories. One of such structures, the double category, is considered in this article. The double category structure is defined as generalization of the bicategory structure. It is shown that double categories exist in the topological and ordinary quantum field theories, and for dynamical systems with inputs and outputs. Morphisms of all these double categories are not maps of sets. 1 Double Category Definition 1.1 A double category D consists of the following: (1) A category D 0 of objects Obj(D 0) and morphisms Mor(D 0) of 0-level. (2) A category D 1 of objects Obj(D 1) of 1-level and morphisms Mor(D 1) of 2-level.