Symmetric Monoidal Categories for Operads - Index of

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244 17 Miscellaneous Applications Nevertheless one can observe that K(M,R, R) (respectively, K(R, R,N)) forms a K-cell complex (respectively, an L-cell complex) in the sense of the definition of §15.3 if M,R,N are connected. Therefore, by theorems 15.3.A-15.3.B, we have at least a weak-equivalence i : K(M,R,N) ∼ −→ B(M,R,N)forconnected objects M,R,N. In [14] we prove this assertion by a spectral sequence argument. 17.2 The Operadic Cotriple Construction The cotriple construction BΔ(P, P,A), whose definition is recalled in §13.3.1, forms a simplicial resolution of A in the sense that the normalized complex of this simplicial P-algebra comes equipped with a weak-equivalence ɛ : N∗(BΔ(P, P,A)) ∼ −→ A. In §13.3 we recall the definition of BΔ(P, P,A) and of the associated normalized complex N∗(BΔ(P, P,A)) in the basic case where A is a P-algebra in dg-modules, but the construction makes sense for a P-algebra in Σ∗-objects, respectively for a P-algebra in right modules over an operad R. Inparticular, we can apply the construction to the P-algebra in right P-modules formed by the operad itself to obtain a simplicial resolution of P in the category of P-algebras in right P-modules. The first purpose of this section is to observe that BΔ(P, P,A) is identified with the functor associated to this universal simplicial resolution: BΔ(P, P,A)=SP(BΔ(P, P, P),A), and similarly as regards the normalized complex N∗(BΔ(P, P,A)). In addition, we review the definition of a cotriple construction with coefficients in a right P-module on a left hand side. The cotriple construction with coefficients satisfies formal identities: BΔ(M,P,A)=SP(BΔ(M,P, P),A)=SP(M,BΔ(P, P,A)) and N∗(BΔ(M,P,A)) = SP(N∗(BΔ(M,P, P)),A), but SP(M,N∗(BΔ(P, P,A))) differs from N∗(BΔ(M,P,A)) = N∗(SP(M,BΔ (P, P,A))) because the functor SP(M,−) does not preserves colimits. Nevertheless, we observe that N∗(BΔ(M,P,A)) determines the evaluation of SP(M,−) on a cofibrant replacement of A, under the assumption that the operad P and the right P-module M are Σ∗-cofibrant objects. For this purpose, we prove that N∗(BΔ(M,P, P)) represents a cofibrant replacement of M in the category of right P-modules and we use the theorems of §15 about the homotopy invariance of functors associated to modules over operads.
17.2 The Operadic Cotriple Construction 245 17.2.1 Cotriple Constructions with Coefficients. The simplicial cotriple construction BΔ(P, P,A)of§13.3.1 is defined for an operad P in any base category C and for a P-algebra A in any category E over C. To form a simplicial cotriple construction with coefficients in a right P-module M, we replace simply the first factor S(P) in the definition of §13.3.1 by the functor S(M) associated to M and we use the morphism BΔ(M,P,A)n =S(M) ◦ S(P) 1 ◦···◦S(P) (A) n S(M) ◦ S(P) =S(M ◦ P) ρ −→ S(M) induced by the right P-action on M to define the face d0 : BΔ(M,P,A)n → BΔ(M,P,A)n−1. The construction returns a simplicial object in E, the underlying category of the P-algebra A. The construction can be applied to a P-algebra N in the category of Σ∗objects E = M, respectively in the category of right modules over an operad E = MR. Inthiscontext,wehaveanidentity BΔ(M,P,N)n = M ◦ P ◦n ◦N since the bifunctor (M,N) ↦→ S(M,N) is identified with the composition product of Σ∗-objects.InthecaseofaP-algebra in right R-modules, the right R-action on BΔ(M,P,N)n is identified with the obvious morphism M ◦ P ◦n ◦N ◦ R M◦P◦n ◦ρ −−−−−−→ M ◦ P ◦n ◦N, where ρ : N ◦ R → N defines the right R-action on N. In the case P = R = N, therightR-action ρ : M ◦ R → M determines a morphism of right R-modules ɛ : BΔ(M,R, R)0 → M so that ɛd0 = ɛd1. Accordingly, the simplicial right R-module BΔ(M,R, R) comes equipped with a natural augmentation ɛ : BΔ(M,R, R) → M, where M is identified with a constant simplicial object. If we also have M = R, then the construction returns a simplicial Ralgebra in right R-modules BΔ(R, R, R) together with an augmentation ɛ : BΔ(R, R, R) → R in the category of R-algebras in right R-modules. In this case, we retrieve the construction of §13.3.1 applied to the R-algebra in right R-modules formed by the operad itself A = R. In the remainder of this section, we assume that the base category is the category of dg-modules C = dgk Mod. In this setting, we can use
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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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140 10 Miscellaneous Examples 10.1.
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