Frobenius algebras and 2D topological quantum field theories

Joachim Kock0521540313, 9780521540315, 0521832675, 9780521832670, 9780511078842

Describing a striking connection between topology and algebra, rather than only proving the theorem, this study demonstrates how the result fits into a more general pattern. Throughout the text emphasis is on the interplay between algebra and topology, with graphical interpretation of algebraic operations, and topological structures described algebraically in terms of generators and relations. Includes numerous exercises and examples.

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Table of contents :
Cover……Page 1
Series-title……Page 3
Title……Page 5
Copyright……Page 6
Dedication……Page 7
Preface……Page 9
Contents……Page 13
General conventions……Page 16
Introduction……Page 17
Summary……Page 25
Manifolds with boundary……Page 26
Orientations……Page 28
Some vocabulary from Morse theory……Page 31
Unoriented cobordisms……Page 34
Oriented cobordisms……Page 38
Decomposition of cobordisms……Page 44
Topological quantum field theories……Page 46
1.3 The category of cobordism classes……Page 50
Gluing and composition……Page 51
Identity cobordisms and invertible cobordisms……Page 60
Monoidal structure……Page 64
Topological quantum field theories……Page 70
Preliminary observations……Page 72
Generators……Page 78
Relations……Page 85
Relations involving the twist……Page 88
Sufficiency of the relations……Page 89
Summary……Page 94
Vector spaces, duals, and pairings……Page 95
Algebras and modules……Page 102
Definition and basic properties……Page 110
Examples……Page 114
2.3 Frobenius algebras and comultiplication……Page 122
Graphical calculus……Page 124
Commutativity and cocommutativity……Page 137
Tensor calculus (linear algebra in coordinates)……Page 139
Frobenius algebra homomorphisms……Page 147
Tensor products of Frobenius algebras……Page 148
Digression on bialgebras……Page 151
Summary……Page 154
Some notions from set theory……Page 155
Definition of monoid……Page 156
Monoid actions and representations……Page 162
3.2 Monoidal categories……Page 164
Definition of monoidal categories……Page 166
Nonstrict monoidal categories……Page 170
Examples of monoidal categories and functors……Page 173
Symmetric monoidal categories……Page 176
Monoidal functor categories……Page 183
3.3 Frobenius algebras and 2-dimensional topological quantum field theories……Page 187
Finite ordinals……Page 193
Graphical description of Delta……Page 196
Generators and relations for Delta……Page 199
The symmetric equivalent: finite cardinals……Page 204
Generators and relations for Phi……Page 208
Monoids in monoidal categories……Page 213
Examples……Page 217
Monoidal functors from the simplex category……Page 220
Algebras……Page 223
Symmetric monoidal functors on Phi……Page 224
Comonoids and coalgebras……Page 228
Frobenius objects, Frobenius algebras, and 2-dimensional cobordisms……Page 230
A.1 Categories……Page 239
A.2 Functors……Page 242
A.3 Universal objects……Page 246
References……Page 250
Index……Page 253