Turaev V., Virelizier A. Monoidal Categories and Topological Field Theory

Springer International Publishing AG, 2017. – 513 p. – (Progress in Mathematics 322) – ISBN: 3319498339.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 center 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). The authors then prove the main result of the monograph: the state sum graph TQFT derived from any spherical fusion category is isomorphic to the Reshetikhin-Turaev surgery graph TQFT derived from the center of that category.
The book is of interest to researchers and students studying topological field theory, monoidal categories, Hopf algebras and Hopf monads.Monoidal Categories
The graphical calculus
Braided categories
Fusion categories
The center of a monoidal category
Hopf Algebras and Monads
Hopf algebras in braided categories
Monads and bimonads
Hopf monads
Monadicity of the center
State Sum Topological Field Theory
Topological Quantum Field Theory
Skeletons of 3-manifolds
Multiplicity modules and colored graphs
The state sum TQFT
Graph Topological Field Theory
Ribbon graphs in 3-manifolds
The state sum graph TQFT
Properties of the state sum graph TQFT
Surgery computation
Appendices
Examples of monoidal categories
Coends
Abelian categories
Hopf monads vs Hopf algebras
Unordered tensor products of modules
The 6j-symbols
Unitary TQFTs
The Dijkgraaf–Witten invariants
Hints and solutions to exercises