Symmetric Monoidal Categories for Operads - Index of

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Benoit Fresse UFR de Mathématiques Université des Sciences et Technologies de Lille Cité Scientifique - Bâtiment M2 59655 Villeneuve d’Ascq Cedex France Benoit.Fresse@math.univ-lille1.fr ISBN: 978-3-540-89055-3 e-ISBN: 978-3-540-89056-0 DOI: 10.1007/978-3-540-89056-0 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008938192 Mathematics Subject Classification (2000): Primary: 18D50; Secondary: 55P48, 18G55, 18A25 c○ Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publisher Services Printed on acid-free paper 987654321 springer.com
Preface The notion of an operad was introduced 40 years ago in algebraic topology in order to model the structure of iterated loop spaces [6, 47, 60]. Since then, operads have been used fruitfully in many fields of mathematics and physics. Indeed, the notion of an operad supplies both a conceptual and effective device to handle a variety of algebraic structures in various situations. Many usual categories of algebras (like the category of commutative and associative algebras, the category of associative algebras, the category of Lie algebras, the category of Poisson algebras, ...) are associated to operads. The main successful applications of operads in algebra occur in deformation theory: the theory of operads unifies the construction of deformation complexes, gives generalizations of powerful methods of rational homotopy, and brings to light deep connections between the cohomology of algebras, the structure of combinatorial polyhedra, the geometry of moduli spaces of surfaces, and conformal field theory. The new proofs of the existence of deformation-quantizations by Kontsevich and Tamarkin bring together all these developments and lead Kontsevich to the fascinating conjecture that the motivic Galois group operates on the space of deformation-quantizations (see [35]). The purpose of this monograph is to study not operads themselves, but modules over operads as a device to model functors between categories of algebras as effectively as operads model categories of algebras. Modules over operads occur naturally when one needs to represent universal complexes associated to algebras over operads (see [14, 54]). Modules over operads have not been studied as extensively as operads yet. However, a generalization of the theory of Hopf algebras to modules over operads has already proved to be useful in various mathematical fields: to organize Hopf invariants in homotopy theory [2]; to study non-commutative generalizations of formal groups [12, 13]; to understand the structure of certain combinatorial Hopf algebras [38, 39]. Besides, the notion of a module over an operad unifies and generalizes classical structures, like Segal’s notion of a v
Page 2 and 3: Lecture Notes in Mathematics 1967 E
Page 6 and 7: vi Preface Γ -object, which occur
Page 8 and 9: viii Contents 9 Algebras in Right M
Page 10 and 11: Introduction Main Ideas and Objecti
Page 12 and 13: Introduction 3 The category of Σ
Page 14 and 15: Introduction 5 the bar module BC to
Page 16 and 17: Introduction 7 modules over envelop
Page 18 and 19: Introduction 9 to a P-algebra in ri
Page 20 and 21: Introduction 11 DerP(A, E). There i
Page 22 and 23: Introduction 13 In both contexts (m
Page 24 and 25: 22 1 Symmetric Monoidal Categories
Page 38 and 39: 36 2 Symmetric Objects and Functors
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52 2 Symmetric Objects and Functors
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54 3 Operads and Algebras in Symmet
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78 4 Miscellaneous Structures Assoc
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Chapter 5 Definitions and Basic Con
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5.1 The Functor Associated to a Rig
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Chapter 6 Tensor Products Introduct
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6.1 The Symmetric Monoidal Category
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6.3 On Enriched Category Structures
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Chapter 7 Universal Constructions o
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7.2 Extension and Restriction of St
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122 8 Adjunction and Embedding Prop
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130 9 Algebras in Right Modules ove
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140 10 Miscellaneous Examples 10.1.
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142 10 Miscellaneous Examples Let
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144 10 Miscellaneous Examples Proof
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146 10 Miscellaneous Examples In th
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Chapter 11 Symmetric Monoidal Model
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11.1 Recollections: The Language of
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11.2 Examples of Model Categories i
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11.3 Symmetric Monoidal Model Categ
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11.4 The Model Category of Σ∗-Ob
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11.4 The Model Category of Σ∗-Ob
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11.5 The Pushout-Product Property f
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11.6 Symmetric Monoidal Categories
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Chapter 12 The Homotopy of Algebras
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12.1 Semi-Model Categories 187 Simi
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12.1 Semi-Model Categories 189 The
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12.1 Semi-Model Categories 191 Say
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12.2 The Semi-Model Category of Ope
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12.2 The Semi-Model Category of Ope
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12.3 The Semi-Model Categories of A
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12.3 The Semi-Model Categories of A
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12.5 The Homotopy of Extension and
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Chapter 13 The (Co)homology of Alge
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13.1 The Construction 205 13.1.2 Pr
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13.1 The Construction 207 associate
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13.2 Universal Coefficient Spectral
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13.3 The Operadic Cotriple Construc
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13.3 The Operadic Cotriple Construc
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Chapter 14 The Model Category of Ri
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14.1 The Symmetric Monoidal Model C
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14.3 Model Categories of Algebras i
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226 15 Modules and Homotopy Invaria
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Chapter 16 Extension and Restrictio
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16.1 Proofs 237 In the remainder of
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16.2 Applications to Bimodules over
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242 17 Miscellaneous Applications c
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244 17 Miscellaneous Applications N
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246 17 Miscellaneous Applications n
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248 17 Miscellaneous Applications B
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250 17 Miscellaneous Applications h
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252 17 Miscellaneous Applications r
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254 17 Miscellaneous Applications N
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256 17 Miscellaneous Applications a
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258 17 Miscellaneous Applications T
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Chapter 18 Shifted Modules over Ope
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18.2 Shifted Functors and Pushouts
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Chapter 19 Shifted Functors and Pus
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19.1 Preliminary Step 279 Proof. By
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19.2 Pushouts 281 In the next proof
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19.2 Pushouts 283 and morphism (1)
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19.3 Third Step: Composites 285 19.
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Chapter 20 Applications of the Push
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20.2 Applications: Homotopy Invaria
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292 References cohomology, and alge
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294 References [57] V. Smirnov, On
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Index adjunction of symmetric monoi
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Index 299 differential graded modul
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Index 301 of simplicial sets, §11.
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Index 303 semi-model category, §12
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Glossary of Notation A: the associa
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Glossary of Notation 307 M 0 : the
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Lecture Notes in Mathematics For in
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Vol. 1874: M. Yor, M. Émery (Eds.)
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LECTURE NOTES IN MATHEMATICS Edited
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