Monoidal Categories and Topological Field Theory - 322

This monograph is devoted to monoidal categories and their connections with 3-dimensional topological field theories.

Starting with basic definitions, it proceeds to the forefront of current research.

Part 1 introduces monoidal categories and several of their classes, including rigid, pivotal, spherical, fusion, braided, and modular categories.

It then presents deep theorems of Müger on the centre of a pivotal fusion category.

These theorems are proved in Part 2 using the theory of Hopf monads.

In Part 3 the authors define the notion of a topological quantum field theory (TQFT) and construct a Turaev-Viro-type 3-dimensional state sum TQFT from a spherical fusion category.

Lastly, in Part 4 this construction is extended to 3-manifolds with colored ribbon graphs, yielding a so-called graph TQFT (and, consequently, a 3-2-1 extended TQFT).