Monoidal Categories and Topological Field Theory

This monograph focuses on monoidal categories and their connection with three-dimensional topological field theories, guiding readers from basic definitions to the forefront of current research.

Part 1 starts by introducing various important classes of monoidal categories, including rigid, pivotal, spherical, fusion and braided categories. It ends by stating two important theorems of M. Müger, establishing fundamental properties of the center of a pivotal fusion category. The theorems are proved in Part 2 using monad, bimonad and Hopf monad techniques. In the third part the authors introduce the notion of topological quantum field theory (TQFT) and formulate the Turaev-Viro-type state sum construction of 3-dimensional TQFTs from spherical fusion categories. Part 4 extends this construction to 3-manifolds with colored ribbon graphs, which results in a so-called graph TQFT. In the last chapter the authors present a surgery computation of the graph TQFT and prove the main result of the monograph: the state sum TQFT derived from a spherical fusion category is isomorphic to the Reshetikhin-Turaev surgery TQFT derived from the center of that category.

The book is of interest to researchers and students studying monoidal categories, Hopf algebras, monads, and 3-dimensional topological field theory.