Symmetric Monoidal Categories for Operads - Index of

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140 10 Miscellaneous Examples 10.1.1 Shifted Operads. Recall that the shifted operad R[ · ] consists of the collection of objects R[m](n) =R(m + n), m, n ∈ N. In §4.1.4, we use the action of Σn on {m +1,...,m+ n} ⊂{1,...,m+ n} to define an action of Σn on R(m + n), for every n ∈ N, and to give to the collection R[m](n), n ∈ N, the structure of a Σ∗-object. In the case of an operad, the composites at last positions R(r + s) ⊗ R(n1) ⊗···⊗R(ns) → R(r + n1 + ···+ ns) (insert operad units η : 1 → R(1)atpositions1,...,r) provide each object R[r], r ∈ N, with the additional structure of a right R-module. In §4.1.5, we also observe that the collection {R[r]}r∈N is equipped with the structure of an operad: the action of Σm on {1,...,m}⊂{1,...,m+ n} determines an action of Σm on R[m](n) =R(m + n), for every m, n ∈ N, and we use operadic composites at first positions R(r + s) ⊗ R(m1 + n1) ⊗···⊗R(mr + nr) → R(m1 + n1 + ···+ mr + nr + s) together with the action of appropriate bloc permutations R(m1 + n1 + ···+ mr + nr + s) � −→ R(m1 + ···+ mr + n1 + ···+ nr + s) to define composition products: R[r] ⊗ R[m1] ⊗···⊗R[mr] → R[m1 + ···+ mr]. The axioms of operads (in May’s form) imply immediately that the action of permutations w ∈ Σm preserves the internal right R-module structure of R[m] and similarly regarding the composition products of R[ · ]. Finally, we obtain that the collection {R[r]}r∈N forms an operad in the symmetric monoidal category of right R-modules. The morphism of §4.1.6 η : R → R[ · ], which identifies R(m) with the constant part of R[m], forms a morphism of operads in right R-modules. In the sequel, we use the notation OR to refer to the category of operads in right R-modules, and the notation R / OR to refer to the comma category of objects under R ∈O, where we identify the objects P(m) underlying an operad P ∈Owith constant right R-modules to form a functor O→OR. Our definition makes the operad R[ · ] an object of this category R/OR. 10.1.2 Functors on Operads. In §4.1.5, we use that S : M ↦→ S(M) defines a functor of symmetric monoidal categories S : M→F to obtain that the collection {S(R[r],X)}r∈N, associated to any object X ∈E, forms an operad in E. We obtain similarly that the collection {SR(R[r],A)}r∈N, associatedto any R-algebra A ∈ RE, forms an operad in E.
10.1 Enveloping Operads and Algebras 141 Furthermore, the morphism η : R → R[ · ] induces a morphism of operads η : R → SR(R[ · ],A) so that SR(R[ · ],A) forms an operad under R. As a conclusion, we obtain that the map SR(R[ · ]) : A ↦→ SR(R[ · ],A) defines a functor SR(R[ · ]) : RE →R / OE. Our goal, announced in the introduction, is to prove: 10.1.3 Theorem. For every R-algebra A ∈ RE, the enveloping operad UR(A) is isomorphic to the operad SR(R[ · ],A) ∈ R / OE associated to A by the functor SR(R[ · ]) : RE →R / OE. Proof. By lemma 4.1.11, the enveloping operad of an R-algebra A is realized by a reflexive coequalizer of the form: R[S(R,A)] φs 0 φd 0 φd 1 R[A] UR(A), where we use the notation R[X] = S(R[ · ],X). To define the morphisms φd0,φd1,φs0 we use that R[X] represents the enveloping operad of the free R-algebra A = R(X) and we apply the adjunction relation of enveloping operads (see §§4.1.8-4.1.11). But a straightforward inspection of constructions shows that these morphisms are identified with the morphisms s0 S(R[ · ], S(R,A)) d0 d1 S(R[ · ],A) which occur in the definition of the functor S(R[ · ],A) associated to the right R-module R[ · ]. Therefore we have an identity UR(A) =SR(R[ · ],A), for every A ∈ RE. ⊓⊔ 10.1.4 Remark. In the point-set context, the relation UR(A) =SR(R[ · ],A) asserts that UR(A) is spanned by elements of the form u(x1,...,xm) =p(x1,...,xm,a1,...,an), where x1,...,xm are variables and a1,...,an ∈ A, together with the relations p(x1,...,xm,a1,...,ae−1,q(ae,...,ae+s−1),ae+s,...,an+s−1) ≡ p ◦m+e q(x1,...,xm,a1,...,ae−1,ae,...,ae+s−1,ae+s,...,an+s−1). Thus we recover the pointwise construction of §4.1.3. 10.1.5 Functoriality of Enveloping Operads. To complete this section, we study the functoriality of enveloping operads.
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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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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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14.1 The Symmetric Monoidal Model C
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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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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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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 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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