Monoidal categories and monoidal 2-categories

Abstract

In this chapter we provide the definitions of various algebraic structures that will be the building blocks for our construction of TQFT functors. In particular, we will review braided abelian tensor categories (BTQ, the properties of Hopf algebras in such BTC’s, as well as the construction of a symmetric monoidal 2-category of abelian categories. Our discussion will also include a number of new lemmas that will significantly simplify the proof of topological invariance in Chapter 6.

More specifically, in the discussion of braided tensor categories we give a detailed discussion of rigidity in BTC’s, the various ribbon and balancing elements, and their relations. As a result, we obtain that any BTC is equivalent to one, which is strictly rigid, i.e., we have X = X ^vv and the canonical balancing is just the identity.