Details
Monoidal Categories and Topological Field Theory
Progress in Mathematics, Band 322
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CHF 212.50 |
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| Verlag: | Birkhäuser |
| Format: | |
| Veröffentl.: | 28.06.2017 |
| ISBN/EAN: | 9783319498348 |
| Sprache: | englisch |
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Beschreibungen
<p>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.</p>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.<p></p><p>The book is of interest to researchers and students studying topological field theory, monoidal categories, Hopf algebras and Hopf monads.</p><p> </p><p> </p>
<p>Introduction.- Part I: Monoidal Categories.- Part 2: Hopf Algebras and Monads.- Part 3: State Sum Topological Field Theory.- Part 4: Graph Topological Field Theory.- Appendices.- Bibliography.- Index.</p>
<p>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.</p><p>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 isomorphicto the Reshetikhin-Turaev surgery graph TQFT derived from the center of that category.</p><p></p><p>The book is of interest to researchers and students studying topological field theory, monoidal categories, Hopf algebras and Hopf monads.</p><p></p>
Offers a detailed exposition accessible to students Provides numerous figures Winner of the 2016 Ferran Sunyer i Balaguer Prize
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