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© 2017

Monoidal Categories and Topological Field Theory

Book

Part of the Progress in Mathematics book series (PM, volume 322)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Monoidal Categories

    1. Front Matter
      Pages 1-1
    2. Vladimir Turaev, Alexis Virelizier
      Pages 3-30
    3. Vladimir Turaev, Alexis Virelizier
      Pages 31-51
    4. Vladimir Turaev, Alexis Virelizier
      Pages 53-63
    5. Vladimir Turaev, Alexis Virelizier
      Pages 65-87
    6. Vladimir Turaev, Alexis Virelizier
      Pages 89-96
  3. Hopf Algebras and Monads

    1. Front Matter
      Pages 97-97
    2. Vladimir Turaev, Alexis Virelizier
      Pages 99-125
    3. Vladimir Turaev, Alexis Virelizier
      Pages 127-155
    4. Vladimir Turaev, Alexis Virelizier
      Pages 157-189
    5. Vladimir Turaev, Alexis Virelizier
      Pages 191-225
  4. State Sum Topological Field Theory

    1. Front Matter
      Pages 227-227
    2. Vladimir Turaev, Alexis Virelizier
      Pages 229-235
    3. Vladimir Turaev, Alexis Virelizier
      Pages 237-255
    4. Vladimir Turaev, Alexis Virelizier
      Pages 257-272
    5. Vladimir Turaev, Alexis Virelizier
      Pages 273-290
  5. Graph Topological Field Theory

    1. Front Matter
      Pages 291-291
    2. Vladimir Turaev, Alexis Virelizier
      Pages 293-319
    3. Vladimir Turaev, Alexis Virelizier
      Pages 321-398

About this book

Introduction

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.

Keywords

monoidal categories Hopf algebras Hopf monads 3-manifold invariants topological quantum field theory state sums

Authors and affiliations

  1. 1.Department of MathematicsIndiana UniversityBloomingtonUSA
  2. 2.Laboratoire Paul PainlevéUniversité de LilleVilleneuve d'Ascq CedexFrance

Bibliographic information

Reviews

“The book gives a self-contained account of the algebraic construction of TQFTs including all required background on the theory of monoidal categories. As such, it appears to be unique in the literature. Material otherwise only accessible through various journal articles, and partly only published in preprints, has been combined into a single well-written and accessible account of the theory. The book combines decades of research into a single text, bringing the reader from the basics to the forefront of research.” (Robert Laugwitz, Mathematical Reviews, July, 2018)