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vi Preface Γ -object, which occur in homological algebra and homotopy theory. In [33], Kapranov and Manin give an application of the relationship between modules over operads and functors for the construction of Morita equivalences between categories of algebras. Our own motivation to write this monograph comes from homotopy theory: we prove, with a view to applications, that functors determined by modules over operads satisfy good homotopy invariance properties. Acknowledgements I am grateful to Geoffrey Powell and the referee for a careful reading of a preliminary version of this work and helpful observations. I thank Clemens Berger, David Chataur, Joan Millès, John E. Harper for useful discussions, and for pointing out many mistakes in the first version of this monograph. Cet ouvrage n’aurait été menéà son terme sans l’atmosphère amicale de l’équipe de topologie de Lille. Je remercie mes collègues, et plus particulièrement David Chataur, Sadok Kallel et Daniel Tanré, pour toutes sortes de soutien durant l’achèvement de ce livre. I thank the referee, the editors and the Springer-Verlag lecture notes team for their efficient processing of the manuscript. The author is supported by the “Laboratoire Paul Painlevé”, UMR 8524 de l’Université des Sciences et Technologie de Lille et du CNRS, by the GDR 2875 “Topologie Algébrique et Applications” du CNRS, and this research is supported in part by ANR grant 06-JCJC-0042 “Opérades, Bigèbres et Théories d’Homotopie”.
Contents Introduction .................................................. 1 Part I Categorical and Operadic Background Foreword: Categorical Conventions ........................... 17 1 Symmetric Monoidal Categories for Operads ............. 21 2 Symmetric Objects and Functors ......................... 35 3 Operads and Algebras in Symmetric Monoidal Categories ................................................ 53 4 Miscellaneous Structures Associated to Algebras over Operads ............................................. 77 Bibliographical Comments on Part I .......................... 95 Part II The Category of Right Modules over Operads and Functors 5 Definitions and Basic Constructions ...................... 99 6 Tensor Products .......................................... 107 7 Universal Constructions on Right Modules over Operads ............................................. 113 8 Adjunction and Embedding Properties ................... 121 vii
Page 2 and 3: Lecture Notes in Mathematics 1967 E
Page 4 and 5: Benoit Fresse UFR de Mathématiques
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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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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