Mathematical Surveys and Monographs Volume 205 Tensor Categories Pavel Etingof Shlomo Gelaki Dmitri Nikshych Victor Ostrik American Mathematical Society Author's final version made available with permission of the publisher, American Mathematical Society. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Tensor Categories Author's final version made available with permission of the publisher, American Mathematical Society. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Author's final version made available with permission of the publisher, American Mathematical Society. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Mathematical Surveys and Monographs Volume 205 Tensor Categories Pavel Etingof Shlomo Gelaki Dmitri Nikshych Victor Ostrik American Mathematical Society Providence, Rhode Island Author's final version made available with permission of the publisher, American Mathematical Society. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms EDITORIAL COMMITTEE Robert Guralnick Constantin Teleman Michael A. Singer, Chair MichaelI.Weinstein Benjamin Sudakov 2010 Mathematics Subject Classification. Primary 17B37, 18D10, 19D23, 20G42. For additional information and updates on this book, visit www.ams.org/bookpages/surv-205 Library of Congress Cataloging-in-Publication Data Tensor categories / Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik. pages cm. — (Mathematical surveys and monographs ; volume 205) Includes bibliographical references and index. ISBN 978-1-4704-2024-6 (alk. paper) 1. Algebraic topology. 2. Tensor fields. 3. Hopf algebras. I. Etingof, P. I. (Pavel I.), 1969– II. Gelaki, Shlomo, 1964– III. Nikshych, Dmitri, 1973– IV. Ostrik, Victor. QA612.T46 2015 512.57—dc23 2015006773 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. 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Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 201918171615 Author's final version made available with permission of the publisher, American Mathematical Society. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Dedicated to our children Miriam and Ariela Etingof Hadar and Klil Gelaki Timofei, Daria, and Nadezhda Nikshych and Tatiana, Valentina, and Yuri Ostrik Author's final version made available with permission of the publisher, American Mathematical Society. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Author's final version made available with permission of the publisher, American Mathematical Society. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms Contents Preface xi Chapter 1. Abelian categories 1 1.1. Categorical prerequisites and notation 1 1.2. Additive categories 1 1.3. Definition of abelian category 2 1.4. Exact sequences 4 1.5. Length of objects and the Jordan-H¨older theorem 5 1.6. Projective and injective objects 6 1.7. Higher Ext groups and group cohomology 7 1.8. Locally finite (artinian) and finite abelian categories 9 1.9. Coalgebras 12 1.10. The Coend construction 14 1.11. Deligne’s tensor product of locally finite abelian categories 15 1.12. The finite dual of an algebra 16 1.13. Pointed coalgebras and the coradical filtration 16 1.14. Bibliographical notes 19 Chapter 2. Monoidal categories 21 2.1. Definition of a monoidal category 21 2.2. Basic properties of unit objects 22 2.3. First examples of monoidal categories 26 2.4. Monoidal functors and their morphisms 30 2.5. Examples of monoidal functors 32 2.6. Monoidal functors between categories of graded vector spaces 33 2.7. Group actions on categories and equivariantization 35 2.8. The Mac Lane strictness theorem 36 2.9. The coherence theorem 39 2.10. Rigid monoidal categories 40 2.11. Invertible objects and Gr-categories 43 2.12. 2-categories 45 2.13. Bibliographical notes 46 Chapter 3. Z+-rings 49 3.1. Definition of a Z+-ring 49 3.2. The Frobenius-Perron theorem 51 3.3. The Frobenius-Perron dimensions 52 3.4. Z+-modules 56 3.5. Graded based rings 58 3.6. The adjoint based subring and universal grading 60 vii Author's final version made available with permission of the publisher, American Mathematical Society. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms viii CONTENTS 3.7. Complexified Z+-rings and ∗-algebras 62 3.8. Weak based rings 63 3.9. Bibliographical notes 63 Chapter 4. Tensor categories 65 4.1. Tensor and multitensor categories 65 4.2. Exactness of the tensor product 66 4.3. Semisimplicity of the unit object 69 4.4. Absence of self-extensions of the unit object 70 4.5. Grothendieck ring and Frobenius-Perron dimension 71 4.6. Deligne’s tensor product of tensor categories 73 4.7. Quantum traces, pivotal and spherical categories 73 4.8. Semisimple multitensor categories 76 4.9. Grothendieck rings of semisimple tensor categories 76 4.10. Categorification of based rings 78 4.11. Tensor subcategories 80 4.12. Chevalley property of tensor categories 81 4.13. Groupoids 82 4.14. The adjoint subcategory and universal grading 83 4.15. Equivariantization of tensor categories 86 4.16. Multitensor categories over arbitrary fields 87 4.17. Bibliographical notes 88 Chapter 5. Representation categories of Hopf algebras 91 5.1. Fiber functors 91 5.2. Bialgebras 91 5.3. Hopf algebras 93 5.4. Reconstruction theory in the infinite setting 97 5.5. More examples of Hopf algebras 99 5.6. The quantum group Uq(sl2) 101 5.7. The quantum group Uq(g) 103 5.8. Representations of quantum groups and quantum function algebras 104 5.9. Absence of primitive elements 106 5.10. The Cartier-Gabriel-Kostant theorem 106 5.11. Pointed tensor categories and Hopf algebras 108 5.12. Quasi-bialgebras 110 5.13. Quasi-bialgebras with an antipode and quasi-Hopf algebras 112 5.14. Twists for bialgebras and Hopf algebras 114 5.15. Bibliographical notes 115 5.16. Other results 117 Chapter 6. Finite tensor categories 119 6.1. Properties of projective objects 119 6.2. Categorical freeness 122 6.3. Injective and surjective tensor functors 124 6.4. The distinguished invertible object 126 6.5. Integrals in quasi-Hopf algebras and unimodular categories 127 6.6. Degeneracy of the Cartan matrix 129 6.7. Bibliographical notes 129 Author's final version made available with permission of the publisher, American Mathematical Society. License or copyright restrictions may apply to redistribution; see http://www.ams.org/publications/ebooks/terms CONTENTS ix Chapter 7. Module categories 131 7.1. The definition of a module category 131 7.2. Module functors 134 7.3. Module categories over multitensor categories 135 7.4. Examples of module categories 136 7.5. Exact module categories over finite tensor categories 138 7.6. First properties of exact module categories 139 7.7. Module categories and Z+-modules 140 7.8. Algebras in multitensor categories 141 7.9. Internal Homs in module categories 147 7.10. Characterization of module categories in terms of algebras 150 7.11. Categories of module functors 154 7.12. Dual tensor categories and categorical Morita equivalence 155 7.13. The center construction 162 7.14. The quantum double construction for Hopf algebras 163 7.15. Yetter-Drinfeld modules 166 7.16. Invariants of categorical Morita equivalence 166 7.17. Duality for tensor functors and Lagrange’s Theorem 170 7.18. Hopf bimodules and the Fundamental Theorem 172 7.19. Radford’s isomorphism for the fourth dual 176 7.20. The canonical Frobenius algebra of a unimodular category 178 7.21. Categorical dimension of a multifusion category 179 7.22. Davydov-Yetter cohomology and deformations of tensor categories 183 7.23. Weak Hopf algebras 186 7.24. Bibliographical notes 189 7.25. Other results 192 Chapter 8. Braided categories 195 8.1. Definition of a braided category 195 8.2. First examples of braided categories and functors 197 8.3. Quasitriangular Hopf algebras 198 8.4. Pre-metric groups and pointed braided fusion categories 203 8.5. The center as a braided category 207 8.6. Factorizable braided tensor categories 208 8.7. Module categories over braided tensor categories 209 8.8. Commutative algebras and central functors 210 8.9. The Drinfeld morphism 214 8.10. Ribbon monoidal categories 216 8.11. Ribbon Hopf algebras 221 8.12. Characterization of Morita equivalence 221 8.13.