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Monoidal Categories and Topological Field Theory

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  • © 2017

Overview

Authors:
  1. Vladimir Turaev

    1. Department of Mathematics, Indiana University, Bloomington, USA

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    You can also search for this author in PubMed   Google Scholar

  2. Alexis Virelizier

    1. Laboratoire Paul Painlevé, Université de Lille, Villeneuve d'Ascq Cedex, France

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    You can also search for this author in PubMed   Google Scholar

  • Offers a detailed exposition accessible to students
  • Provides numerous figures
  • Winner of the 2016 Ferran Sunyer i Balaguer Prize

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

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xii
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  2. Monoidal Categories

    1. Front Matter

      Pages 1-1
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    2. Monoidal categories and functors

      • Vladimir Turaev, Alexis Virelizier
      Pages 3-30

    3. The graphical calculus

      • Vladimir Turaev, Alexis Virelizier
      Pages 31-51

    4. Braided categories

      • Vladimir Turaev, Alexis Virelizier
      Pages 53-63

    5. Fusion categories

      • Vladimir Turaev, Alexis Virelizier
      Pages 65-87

    6. The center of a monoidal category

      • Vladimir Turaev, Alexis Virelizier
      Pages 89-96

  3. Hopf Algebras and Monads

    1. Front Matter

      Pages 97-97
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    2. Hopf algebras in braided categories

      • Vladimir Turaev, Alexis Virelizier
      Pages 99-125

    3. Monads and bimonads

      • Vladimir Turaev, Alexis Virelizier
      Pages 127-155

    4. Hopf monads

      • Vladimir Turaev, Alexis Virelizier
      Pages 157-189

    5. Monadicity of the center

      • Vladimir Turaev, Alexis Virelizier
      Pages 191-225

  4. State Sum Topological Field Theory

    1. Front Matter

      Pages 227-227
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    2. Topological Quantum Field Theory

      • Vladimir Turaev, Alexis Virelizier
      Pages 229-235

    3. Skeletons of 3-manifolds

      • Vladimir Turaev, Alexis Virelizier
      Pages 237-255

    4. Multiplicity modules and colored graphs

      • Vladimir Turaev, Alexis Virelizier
      Pages 257-272

    5. The state sum TQFT

      • Vladimir Turaev, Alexis Virelizier
      Pages 273-290

  5. Graph Topological Field Theory

    1. Front Matter

      Pages 291-291
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    2. Ribbon graphs in 3-manifolds

      • Vladimir Turaev, Alexis Virelizier
      Pages 293-319

    3. The state sum graph TQFT

      • Vladimir Turaev, Alexis Virelizier
      Pages 321-398
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Keywords

About this book

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 tothe 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.

 

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)

Authors and Affiliations

  • Department of Mathematics, Indiana University, Bloomington, USA

    Vladimir Turaev

  • Laboratoire Paul Painlevé, Université de Lille, Villeneuve d'Ascq Cedex, France

    Alexis Virelizier

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