Construction of TQFT-Double Functors

Abstract

In Sect. 6.1 of this chapter we give the precise definition of an extended TQFT and state the main result of this book, namely, the existence of a large class of such TQFT’s. Then we proceed with the construction of a TQFT from a given modular category. For this we define some functors, representing surfaces with holes, as coends of expressions involving tensor products, the Hopf algebra F and the functor of invariants. In a sequence of lemmas we show that a cobordism between two surfaces determines a natural transformation between the corresponding functors. To achieve this we first associate a natural transformation to an equivalence class of tangles under ambient isotopy. Then we show that it depends on a wider equivalence class, stable under topological moves, which defines the category of cobordisms as a quotient of the category of tangles (Theorem 3.0.6).