Symmetric Monoidal Categories for Operads - Index of

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128 8 Adjunction and Embedding Properties We apply this result to the generating objects M = Fr ◦ R = R ⊗r .Weform a commutative diagram: HomM R (Fr ◦ R,N) SR HomF R (SR(Fr ◦ R), SR(N)) � HomF R (S(Fr) ◦ U, SR(N)) � � HomF R (U ⊗r , SR(N)) � N � � � HomF (S(Fr), SR(N) ◦ R(−)) � HomF (Id ⊗r , SR(N) ◦ R(−)) HomM(Fr,N) S HomF (S(Fr), S(N)) � � HomF (Id ⊗r , S(N)) By proposition 8.2.2, the left-hand side composite represents the adjunction unit ηR(N) :N(r) → ΓR(SR(N)). By proposition 2.3.8, the right-hand side composite represents the adjunction unit η(N) :N(r) → Γ(S(N)). The bottom isomorphisms represent the isomorphisms ΓR(SR(N)) � Γ(SR(N) ◦ R(−)) � Γ(S(N)) of lemma 8.3.1 and observation 8.3.2. Hence the commutativity of the diagram implies lemma 8.3.3. ⊓⊔ The conclusion of theorem 8.A is an immediate consequence of this lemma and of proposition 2.3.12. ⊓⊔ � .
Chapter 9 Algebras in Right Modules over Operads Introduction The purpose of this chapter is to survey applications of the general statements of §3 to the category of right modules over an operad R. In §2, we prove that the functor S : M ↦→ S(M) defines a functor of symmetric monoidal categories over C S:(M, ⊗, 1) → (F, ⊗, 1) and we use this statement in §3.2 in order to model functors F : E→ PE, where P is an operad in C, byP-algebras in the category of Σ∗-objects. In a similar way, we can use that the functor SR : M ↦→ SR(M) defines a functor of symmetric monoidal categories over C SR :(M R, ⊗, 1) → (F R, ⊗, 1), to model functors F : RE → PE by P-algebras in the category of right R-modules. In this chapter, we also check that the structure of a P-algebra in right R-modules, for P an operad in C, is equivalent to the structure of a P-R-bimodule, an object equipped with both a left P-action and a right R-action in the monoidal category (M, ◦,I), and we give an analogue of our constructions in the bimodule formalism. In the sequel, we aim to extend results of the theory of algebras over operads. For that reason we use the language of algebras in symmetric monoidal categories rather than the bimodule language. B. Fresse, Modules over Operads and Functors, Lecture Notes in Mathematics 1967, 129 DOI: 10.1007/978-3-540-89056-0 9, c○ Springer-Verlag Berlin Heidelberg 2009
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Lecture Notes in Mathematics 1967 E
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Benoit Fresse UFR de Mathématiques
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vi Preface Γ -object, which occur
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viii Contents 9 Algebras in Right M
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Introduction Main Ideas and Objecti
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Introduction 3 The category of Σ
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Introduction 5 the bar module BC to
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Introduction 7 modules over envelop
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Introduction 9 to a P-algebra in ri
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Introduction 11 DerP(A, E). There i
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Introduction 13 In both contexts (m
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22 1 Symmetric Monoidal Categories
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36 2 Symmetric Objects and Functors
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54 3 Operads and Algebras in Symmet
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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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