Generalization of a modular functor

Abstract

In this chapter we are defining enhanced double categories $\widetilde{Cob}^ \cap$ which are equivalent to Cob but have larger sets of 1-arrows. These 1-arrows, called arc-diagrams, encode the structure of certain coends. The reason to introduce them is to define horizontal compositions that do not require braidings. Hence, it will be also be possible to define a TQFT functor with only canonical isomorphisms. The construction is described in Section 7.3. It requires a modification of the double category of tangles to reflect the modifications in the enhanced cobordism-category.

We describe how the braided Hopf algebra F and its integrals look like in familiar examples. Specifically, we consider the case of a semisimple monoidal category as well as the example of the category of modules of a linear ribbon Hopf algebras in the original sense.