Symmetric Monoidal Categories for Operads - Index of

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13.1 The Construction 205 13.1.2 Proposition (see [26]). The functor B ↦→ DerP(B,E) preserves weak-equivalences between cofibrant objects in the category of P-algebras over A. The proof of this proposition is postponed to the end of this section, assuming a general result, proved in the next part, on the homotopy invariance of functors associated to right modules over operads. 13.1.3 Kähler Differentials and Homology. To define the homology of Palgebras, we use the module of Kähler differentials Ω1 P (A) andthecoefficients consist of right modules over the enveloping algebra UP(A). These objects are defined in §4 for operads and algebras in k-modules. The principle of generalized point tensors can be used to extend the construction to the dgcontext. Then the definition returns a dg-algebra UP(A) and a dg-module Ω1 P(A) so that we have an isomorphism DerP(A, E) � Hom UP(A)Mod(Ω 1 P (A),E) in the category of dg-modules. The homology of A with coefficients in a right UP(A)-module E is defined by: H P ∗ (A, E) =H∗(E ⊗ UP(QA) Ω 1 P (QA)), where, again, QA is any cofibrant replacement of A in the category of Palgebras. The next proposition ensures, as in the context of the cohomology, that this definition makes sense: 13.1.4 Proposition (see [26]). The functor B ↦→ E⊗UP(B)Ω 1 P (B) preserves weak-equivalences between cofibrant objects in the category of P-algebras over A. The proof of this lemma is also postponed to the end of this section. 13.1.5 Classical Examples. The definition of the cohomology H∗ P (A, E), respectively homology HP ∗(A, E), can be applied to the usual operads P = A, L, C, provided that the ground ring k has characteristic 0 in the case P = L, C (since these operads are not Σ∗-cofibrant in positive characteristic). The cohomology H∗ P (A, E), respectively homology HP ∗(A, E), defined by the theory of operads agrees with: – the classical Hochschild cohomology, respectively homology, in the case of the associative operad P = A, – the Chevalley-Eilenberg cohomology, respectively homology, in the case of the Lie operad P = L, – the Harrison cohomology, respectively homology, in the commutative operad P = C. These assertions are consequences of results of [17, 18] (see [46]). The idea is to use particular cofibrant replacements, the Koszul resolution, and to identify the dg-module DerP(QA,E), respectively E ⊗ UP(QA) Ω 1 (QA), associated to
206 13 The (Co)homology of Algebras over Operads this resolution with the usual Hochschild (respectively, Chevalley-Eilenberg, Harrison) complex. The Hochschild cohomology and the Chevalley-Eilenberg cohomology (respectively, homology) can also be defined by an Ext-functor (respectively, a Tor-functor). This result has an interpretation in terms of a general universal coefficient spectral sequence, defined in the next section, which relates any cohomology (respectively, homology) associated to an operad to Ext-functors (respectively, Tor-functors). 13.1.6 ¶ Remark. If we use simplicial k-modules rather than dg-modules, then the definition of the cohomology H ∗ P (A, E), respectively homology H P ∗(A, E), makes sense for any simplicial operad P which is cofibrant in simplicial k-modules, and not only for Σ∗-cofibrant operads. In particular, we can use P-algebras in simplicial k-modules to associate a well-defined (co)homology theory to the Lie, respectively commutative, operad in positive characteristic. In the case of the Lie operad, this simplicial (co)homology agrees again with the Chevalley-Eilenberg (co)homology. In the case of the commutative operad, this simplicial (co)homology agrees with André-Quillen’ (co)homology. 13.1.7 Universal Coefficients. The remainder of the section is devoted to the proof of proposition 13.1.2 and proposition 13.1.4, the homotopy invariance of the dg-modules DerP(QA,E)andE ⊗ UP(QA) Ω 1 P (QA), for QA a cofibrant replacement of A. For this aim, we use the representation T 1 P (QA) =UP(A) ⊗ UP(QA) Ω 1 P (QA) which determines the homology of A with coefficients in the enveloping algebra UP(A). The enveloping algebra UP(A) gives universal coefficients for the homology of A since we have identities DerP(QA,E)=HomUP(QA)Mod(Ω 1 P (QA),E) = HomUP(A)Mod(T 1 P (QA),E) and E ⊗UP(QA) Ω 1 P (QA) =E ⊗UP(A) T 1 P (QA). The idea is to prove that the functor QA ↦→ T 1 P (QA) maps cofibrant P-algebras over A to cofibrant left UP(A)-modules and preserves weakequivalences between cofibrant P-algebras. Then we apply the classical homotopy theory of modules over dg-algebras to conclude that the functors QA ↦→ Hom UP(A)Mod(T 1 P (QA),E) and QA ↦→ E ⊗ UP(A) T 1 P (QA) preserve weak-equivalences between cofibrant P-algebras over A. In our verifications, we use repeatedly that the extension and restriction functors φ! : R Mod ⇄ S Mod : φ ∗
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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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5.1 The Functor Associated to a Rig
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6.1 The Symmetric Monoidal Category
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140 10 Miscellaneous Examples 10.1.
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19.2 Pushouts 283 and morphism (1)
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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 299 differential graded modul
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