Frobenius Algebras and 2-D Topological Quantum Field Theories (London Mathematical Society Student Texts, Series Number 59)

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. 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 Cover 1 Series-title 3 Title 5 Copyright 6 Dedication 7 Preface 9 Contents 13 General conventions 16 Introduction 17 1 Cobordisms and topological quantum field theories 25 Summary 25 1.1 Geometric preliminaries 26 Manifolds with boundary 26 Orientations 28 Some vocabulary from Morse theory 31 1.2 Cobordisms 34 Unoriented cobordisms 34 Oriented cobordisms 38 Decomposition of cobordisms 44 Topological quantum field theories 46 1.3 The category of cobordism classes 50 Gluing and composition 51 Identity cobordisms and invertible cobordisms 60 Monoidal structure 64 Topological quantum field theories 70 1.4 Generators and relations for 2Cob 72 Preliminary observations 72 Generators 78 Relations 85 Relations involving the twist 88 Sufficiency of the relations 89 2 Frobenius algebras 94 Summary 94 2.1 Algebraic preliminaries 95 Vector spaces, duals, and pairings 95 Algebras and modules 102 2.2 Definition and examples of Frobenius algebras 110 Definition and basic properties 110 Examples 114 2.3 Frobenius algebras and comultiplication 122 Graphical calculus 124 Commutativity and cocommutativity 137 Tensor calculus (linear algebra in coordinates) 139 2.4 The category of Frobenius algebras 147 Frobenius algebra homomorphisms 147 Tensor products of Frobenius algebras 148 Digression on bialgebras 151 3 Monoids and monoidal categories 154 Summary 154 3.1 Monoids (in Set) 155 Some notions from set theory 155 Definition of monoid 156 Monoid actions and representations 162 3.2 Monoidal categories 164 Definition of monoidal categories 166 Nonstrict monoidal categories 170 Examples of monoidal categories and functors 173 Symmetric monoidal categories 176 Monoidal functor categories 183 3.3 Frobenius algebras and 2-dimensional topological quantum field theories 187 3.4 The simplex categories Delta and Phi 193 Finite ordinals 193 Graphical description of Delta 196 Generators and relations for Delta 199 The symmetric equivalent: finite cardinals 204 Generators and relations for Phi 208 3.5 Monoids in monoidal categories, and monoidal functors from Delta 213 Monoids in monoidal categories 213 Examples 217 Monoidal functors from the simplex category 220 Algebras 223 Symmetric monoidal functors on Phi 224 3.6 Frobenius structures 228 Comonoids and coalgebras 228 Frobenius objects, Frobenius algebras, and 2-dimensional cobordisms 230 Appendix: vocabulary from category theory 239 A.1 Categories 239 A.2 Functors 242 A.3 Universal objects 246 References 250 Index 253 This 2003 book describes a striking connection between topology and algebra, namely that 2D topological quantum field theories are equivalent to commutative Frobenius algebras. The precise formulation of the theorem and its proof is given in terms of monoidal categories, and the main purpose of the book is to develop these concepts from an elementary level, and more generally serve as an introduction to categorical viewpoints in mathematics. Rather than just proving the theorem, it is shown how the result fits into a more general pattern concerning universal monoidal categories for algebraic structures. Throughout, the 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. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques that will prove useful for future work. "Throughout, the 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. The book will prove valuable to students or researchers entering the new quantum areas of mathematics, who will learn a host of modern techniques that will prove useful for future work."--Page 4 de la couverture "Throughout, the 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. The book will prove valuable to students or researchers entering the new quantum areas of mathematics, who will learn a host of modern techniques that will prove useful for future work."--Jacket Proves striking results connecting topology and algebra and shows how the result fits into a more general pattern. The book will prove valuable to students or researchers entering this field who will learn a host of modern techniques. There are numerous exercises and examples making the book suitable for teaching In the first section we recall some basic notions of manifolds with boundary and orientations, and Morse functions.