Cayley-Klein topological quantum field theory

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 Cayley-Klein double category, is considered in this talk. The Cayley-Klein double category structure is defined as generalization of the Cayley-Klein bicategory structure. It is shown that Cayley-Klein double categories exist in the topological quantum field theory.