## Iterated bar complexes of E-infinity algebras and homology theories (2008)

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Citations: | 5 - 2 self |

### BibTeX

@MISC{Fresse08iteratedbar,

author = {Benoit Fresse},

title = {Iterated bar complexes of E-infinity algebras and homology theories },

year = {2008}

}

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### Abstract

We proved in a previous article that the bar complex of an E ∞-algebra inherits a natural E ∞-algebra structure. As a consequence, a well-defined iterated bar construction B n (A) can be associated to any algebra over an E ∞-operad. In the case of a commutative algebra A, our iterated bar construction reduces to the standard iterated bar complex of A. The first purpose of this paper is to give a direct effective definition of the iterated bar complexes of E ∞-algebras. We use this effective definition to prove that the n-fold bar construction admits an extension to categories of algebras over En-operads. Then we prove that the n-fold bar complex determines the homology theory associated to the category of algebras over an En-operad. In the case n = ∞, we obtain an isomorphism between the homology of an infinite bar construction and the usual Γ-homology with trivial coefficients.

### Citations

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Citation Context ... independently from our work, by relying, at some point, on the relationship between topological little n-cubes operads �� B n �� E C �� �� C B n�� � �� 6 Benoit Fresse and iterated loop spaces (see =-=[12, 41]-=-). In the dg-context, we follow another, more direct, approach. First, to define the iterated bar complex of En-algebras, we observe (Theorem 5.4) that the twisting homomorphism ∂E : T n ◦ E → T n ◦ E... |

230 |
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Citation Context ...-modules M is equivalent to the category of functors F : Bij → C , where Bij refers to the category formed by finite sets as objects and bijective maps as morphisms (this equivalence is borrowed from =-=[26, 28]-=-, see also the surveys of [18, §1.1.8] and [40, §1.7]). In one direction, for a functor F : Bij → C , the dg-module F({1, . . . , r}) associated to the set e = {1, . . . , r} inherits an action of the... |

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Citation Context ... independently from our work, by relying, at some point, on the relationship between topological little n-cubes operads �� B n �� E C �� �� C B n�� � �� 6 Benoit Fresse and iterated loop spaces (see =-=[12, 41]-=-). In the dg-context, we follow another, more direct, approach. First, to define the iterated bar complex of En-algebras, we observe (Theorem 5.4) that the twisting homomorphism ∂E : T n ◦ E → T n ◦ E... |

162 | Bourbaki, N., Groupes et algèbres de - Bourbaki, Générale, et al. - 1960 |

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Citation Context ...-modules M is equivalent to the category of functors F : Bij → C , where Bij refers to the category formed by finite sets as objects and bijective maps as morphisms (this equivalence is borrowed from =-=[26, 28]-=-, see also the surveys of [18, §1.1.8] and [40, §1.7]). In one direction, for a functor F : Bij → C , the dg-module F({1, . . . , r}) associated to the set e = {1, . . . , r} inherits an action of the... |

116 | Operads and motives in deformation quantization
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Citation Context ...new nature on the cohomology of iterated loop spaces – notably, our result implies that certain groups of homotopy automorphisms of En-operads act on this cohomology and we conceive, from insights of =-=[32]-=-, that this gives an action of higher versions of the Grothendieck-Teichmüller group on the cohomology of iterated loop spaces. Besides, for a sphere X = Sn−m , we have a chain complex, defined purely... |

116 | Operads in algebra, topology and physics - Markl, Shnider, et al. - 2002 |

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51 | Combinatorial operad actions on cochains
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Citation Context ...torial E∞-operad E equipped with a filtration of the form (*). In fact, we will observe that this E∞-operad has all extra structures needed for the constructions of the article. Besides, we proved in =-=[10]-=- that the Barratt-Eccles acts on cochain complexes of spaces. Thus the Barratt-Eccles operad is also well suited for the topological applications of our results.8 Benoit Fresse Plan This paper is in ... |

45 |
The homology of Cn+1-spaces, n ≥ 0. In The homology of iterated loop spaces., volume 533
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Citation Context ...) is the iterated bar complex of the composition unit I ∈ M, viewed as a commutative algebra equipped with a trivial structure. We focus on the case n > 1. The homology operad H∗(En) is determined in =-=[16]-=- and has a nice description as a composite of the commutative operad and a desuspension of the Lie operad. This composite is usually called the Gerstenhaber operad and is denoted by Gn. The Gerstenhab... |

32 | The Eckmann-Hilton argument, higher operads and En-spaces
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Citation Context ...] (see also [25]), that the twisting homomorphism ∂E, obtained by our lifting construction, is formally equivalent to a twisting cochain on a poset formed from Batanin’s category of pruned trees (see =-=[6, 7]-=-) with values in the operad E. Throughout the article, we study the application of our constructions to the BarrattEccles operad, a nice combinatorial E∞-operad E equipped with a filtration of the for... |

30 |
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Citation Context ...ative algebra, for every ground ring k. Thus our result relates the calculation of TorΣr ∗ (Λ−1 L ∨ (r), k) to classical homological computations. Such Tor-functors are determined by other methods in =-=[2, 3]-=- in the case k = Fp (see also [11] for another approach to this computation). Afterword: applications to the cohomology of iterated loop spaces The goal of this concluding part is to explain the appli... |

28 |
Homotopy limits of homotopy algebras
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Citation Context ...f our main theorems to the cohomology of iterated loop spaces.Iterated bar complexes of E-infinity algebras 93 Let ¯N ∗ (X) denote to the reduced normalized cochain complex of a simplicial set X. By =-=[30]-=- (see also [10, 42]), the cochain complex ¯N ∗ (X) inherits an action of an E∞operad E so that the map N∗ : X ↦→ ¯N ∗ (X) defines a functor from the category of simplicial sets S to the category of E-... |

26 | Combinatorial models for real configuration spaces and En-operads - Berger - 1995 |

26 | Distributive laws and Koszulness
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Citation Context ...on as a composite of the commutative operad and a desuspension of the Lie operad. This composite is usually called the Gerstenhaber operad and is denoted by Gn. The Gerstenhaber operad is Koszul (see =-=[27, 38]-=-) and this property gives the deep reason for the acyclicity of the iterated bar module B n En . For technical reasons, we split the proof of the acyclicity of E 1 = H∗(B n (I)) ◦ H∗(En) in two steps ... |

26 | Multivariable cochain operations and little n-cubes. math.QA/010602
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Citation Context ...rems to the cohomology of iterated loop spaces.Iterated bar complexes of E-infinity algebras 93 Let ¯N ∗ (X) denote to the reduced normalized cochain complex of a simplicial set X. By [30] (see also =-=[10, 42]-=-), the cochain complex ¯N ∗ (X) inherits an action of an E∞operad E so that the map N∗ : X ↦→ ¯N ∗ (X) defines a functor from the category of simplicial sets S to the category of E-algebras in dg-modu... |

24 |
Commutative algebras and cohomology
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Citation Context ...m: Indec B(A) = (Indec T c (ΣA), ∂) = (L c (ΣA), ∂). The chain complex (L c (ΣA), ∂) is a generalization, in the context of Σ∗-modules, of the standard Harrison complex with trivial coefficients (see =-=[29]-=-). The next Proposition is classical for the standard Harrison complex of a free commutative algebra over a field of characteristic zero. 6.9 Proposition The Harrison complex of the free commutative a... |

20 |
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Citation Context ...ative algebras. C E � � 1.4 The example of the Barratt-Eccles operad The Barratt-Eccles operad is a classical instance of an E∞-operad introduced by M. Barratt and P. Eccles in the simplicial setting =-=[4]-=-. Throughout the paper, we use this nice combinatorial operad to illustrate our constructions. For our purpose, we consider a dg-version of the Barratt-Eccles operad which is defined by the normalized... |

17 |
Opérades cellulaires et espaces de lacets itérés, Ann. Inst. Fourier 46
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Citation Context ...s of E-infinity algebras 47 3 Interlude: operads shaped on complete graph posets For our analysis of iterated bar modules, we are going to use a particular cell structure of E∞-operads, introduced in =-=[9]-=-, and modeled by a certain operad in posets K, the complete graph operad. The main purpose of this section is to revisit definitions of [9] in order to give an abstract formalization of the complete g... |

17 | Groupes et algèbres de Lie, Chapitres 2 et 3 - Bourbaki - 1972 |

15 | The symmetrisation of n-operads and compactification of real configuration spaces
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Citation Context ...] (see also [25]), that the twisting homomorphism ∂E, obtained by our lifting construction, is formally equivalent to a twisting cochain on a poset formed from Batanin’s category of pruned trees (see =-=[6, 7]-=-) with values in the operad E. Throughout the article, we study the application of our constructions to the BarrattEccles operad, a nice combinatorial E∞-operad E equipped with a filtration of the for... |

12 |
Gamma homology, Lie representations and
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Citation Context ...o associate an infinite bar complex Σ −∞ B ∞ (A) = colimn Σ −n B n (A) to any E-algebra A. The relationship H En ∗ (A) = H∗(Σ −n B n (A)) also holds in the case n = ∞ (Theorem 9.5). The Γ-homology of =-=[45]-=- and the E∞ André-Quillen homology of [37] are other definitions of the homology theory associated to an E∞-operad. Our result implies that the Γ-homology with trivial coefficient agrees with the homo... |

11 | Product and other fine structure in polynomial resolutions of mapping spaces
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Citation Context ...r n > 1, an associative algebra for n = 1). In the case of the cochain algebra A = C ∗ (X) of a space X, we conjecture that this spectral sequence agrees, from E 2 -stage, with a spectral sequence of =-=[1]-=- which is defined with Goodwillie’s calculus of functors. On one hand, one might gain quantitative information on the cohomology of iterated loop spaces from the study of such spectral sequences, aris... |

11 | On the homotopy of simplicial algebras over an operad - Fresse |

10 |
Homologie et modèle minimal des algèbres de
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Citation Context ...on as a composite of the commutative operad and a desuspension of the Lie operad. This composite is usually called the Gerstenhaber operad and is denoted by Gn. The Gerstenhaber operad is Koszul (see =-=[27, 38]-=-) and this property gives the deep reason for the acyclicity of the iterated bar module B n En . For technical reasons, we split the proof of the acyclicity of E 1 = H∗(B n (I)) ◦ H∗(En) in two steps ... |

8 |
Topological Andre-Quillen cohomology and E∞ Andrew-Quillen cohomology
- Mandell
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Citation Context ... ∞ (A) = colimn Σ −n B n (A) to any E-algebra A. The relationship H En ∗ (A) = H∗(Σ −n B n (A)) also holds in the case n = ∞ (Theorem 9.5). The Γ-homology of [45] and the E∞ André-Quillen homology of =-=[37]-=- are other definitions of the homology theory associated to an E∞-operad. Our result implies that the Γ-homology with trivial coefficient agrees with the homology of the infinite bar complex Σ −∞ B ∞ ... |

6 |
duality of operads and homology of partition posets, in “Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic
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Citation Context ...r of the morphisms d0, d1 : M ◦ R ◦N ⇒ M ◦ N induced by the right R-action on M and the left R-action on N (see for instance [18, §5.1.5, §9.2.4] or [20, §2.1.7], we also refer to the bibliography of =-=[18]-=- for further references on this definition). The extension and restriction functors associated to an operad morphism φ : P → Q, respectively ψ : R → S, are denoted by: φ! : P M ⇄ Q M : φ ∗ , respectiv... |

5 | The homology of certain subgroups of the symmetric group with coefficients in
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- 1998
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Citation Context ...ative algebra, for every ground ring k. Thus our result relates the calculation of TorΣr ∗ (Λ−1 L ∨ (r), k) to classical homological computations. Such Tor-functors are determined by other methods in =-=[2, 3]-=- in the case k = Fp (see also [11] for another approach to this computation). Afterword: applications to the cohomology of iterated loop spaces The goal of this concluding part is to explain the appli... |

5 | Cyclic homology, Die Grundlehren der Mathematischen Wissenschaften 301 - Loday - 1998 |

4 |
over operads and functors
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Citation Context ... E∞-operads. Roughly, we show that the definition of the iterated bar complex B n (A) can be reduced to a construction of linear homological algebra in the context of operads. Let R be any operad. In =-=[20]-=-, we show that a functor SR(M, −) : R C → C is naturally associated to any right R-module M and all functors on R-algebras which are defined by composites of colimits and tensor products have this for... |

4 |
On the chain complex of an iterated loop space, Izv
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- 1989
(Show Context)
Citation Context ...e case k = Q (assuming in this case that X is n-connected). The explicit construction of the n-fold bar complex implies the existence of a spectral sequence B n (H ∗ (X)) ⇒ H ∗ (B n (X)). We refer to =-=[46, 47]-=- for another definition of a similar spectral sequence converging to H∗(Ω n X). We conjecture that our spectral sequence is isomorphic (from the E 2 -stage and up to k-duality) to the H∗(−, k)-version... |

3 |
New approach to the groups H∗(Σn, Lien) by the homology theory of the category of functors
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Citation Context ... k. Thus our result relates the calculation of TorΣr ∗ (Λ−1 L ∨ (r), k) to classical homological computations. Such Tor-functors are determined by other methods in [2, 3] in the case k = Fp (see also =-=[11]-=- for another approach to this computation). Afterword: applications to the cohomology of iterated loop spaces The goal of this concluding part is to explain the applications of our main theorems to th... |

3 | catégorie des arbres élagués de Batanin est de Koszul, preprint arXiv:0909.5447 - La - 2009 |

3 |
The equivalence of certain categories of twisted and Hopf algebras over a commutative ring
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Citation Context ... Benoit Fresse The classical Milnor-Moore and Poincaré-Birkhoff-Witt theorems, which give the structure of cocommutative Hopf algebras, have a natural generalization in the context of Σ∗-modules (see =-=[48]-=-) and so do the dual statements which apply to commutative Hopf algebras. If we restrict ourselves to Σ∗-objects such that M(0) = 0, then the generalized Milnor-Moore and Poincaré-Birkhoff-Witt theore... |

2 |
et al., Algèbres d’Eilenberg-Mac Lane et homotopie, Séminaire Henri Cartan
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Citation Context ... The identity H En ∗ (A) = H∗(Σ −n B n (A)) enables us to deduce the En-homology of usual commutative algebras (polynomial algebras, exterior algebras, divided power algebras, . . . ) from results of =-=[14]-=-. In the case n = ∞, this approach could be used to give explicit representatives of Γ-homology classes and to improve on results of [44].Iterated bar complexes of E-infinity algebras 7 In general, w... |

2 | Adem-Cartan operads - Chataur, Livernet |

2 |
duality of En-operads
- Koszul
(Show Context)
Citation Context ...sted sequence of operads En(r) = colimκ∈Kn(r) E(κ), formed from the Barratt-Eccles operad E, and weakly-equivalent to the nested sequence of the chain operads of little n-cubes (see [9]). We refer to =-=[23]-=- for details about the cofibration condition (K1) in the dg-setting. We just recall the proof of the acyclicity condition (K2). We arrange the definition of the standard contracting chain homotopy ν :... |

2 |
duality complexes for the cohomology of iterated loop spaces of spheres, preprint arXiv:1001.4650
- Koszul
- 2010
(Show Context)
Citation Context ...op spaces. Besides, for a sphere X = Sn−m , we have a chain complex, defined purely algebraically (in terms of characteristic structures of En-operads), computing the cohomology HEn ∗ (C∗(Sn−m)) (see =-=[24]-=-). Thus spheres are first examples of spaces for which our approach seems appropriate and for which we plan to study the applications of our results. Since the publication of this preprint, an interpr... |

2 | An interpretation of En-homology as functor homology
- Livernet, Richter
(Show Context)
Citation Context ... study the applications of our results. Since the publication of this preprint, an interpretation of the homology of iterated bar complexes in terms of a homology of functors has been discovered (see =-=[21, 33]-=-). This new connection gives an additional approach for the study of the homology of En-algebras by means of iterated bar complexes. Notably, we explain in the followup [21] (see also [25]), that the ... |

2 |
A lower bound for coherences on the Brown-Peterson spectrum
- Richter
(Show Context)
Citation Context ...rior algebras, divided power algebras, . . . ) from results of [14]. In the case n = ∞, this approach could be used to give explicit representatives of Γ-homology classes and to improve on results of =-=[44]-=-.Iterated bar complexes of E-infinity algebras 7 In general, we have a natural spectral sequence E 1 = Σ −n B n (H∗(A)) ⇒ H∗(Σ −n B n (A)) = H En ∗ (A), whose E 1 -term reduces to the usual n-fold ba... |

1 |
Homology of En ring spectra and iterated THH, preprint arXiv:1007.5315
- Basterra, Mandell
- 2010
(Show Context)
Citation Context ...ned abstractly as the homology of a derived indecomposable functor L Indec : Ho(En C) → Ho(C). Since the publication of the present preprint, a topological version of this result has been obtained in =-=[5]-=-, independently from our work, by relying, at some point, on the relationship between topological little n-cubes operads �� B n �� E C �� �� C B n�� � �� 6 Benoit Fresse and iterated loop spaces (see... |

1 |
bar complexes and the poset of pruned trees, addendum available at ../~fresse/IteratedBarAppendix.html
- Iterated
(Show Context)
Citation Context ...ed (see [21, 33]). This new connection gives an additional approach for the study of the homology of En-algebras by means of iterated bar complexes. Notably, we explain in the followup [21] (see also =-=[25]-=-), that the twisting homomorphism ∂E, obtained by our lifting construction, is formally equivalent to a twisting cochain on a poset formed from Batanin’s category of pruned trees (see [6, 7]) with val... |

1 | bialgebras and triples of operads, Astérisque 320, Société Mathématique de - Generalized - 2008 |

1 |
duality of En-operads, preprint
- Koszul
(Show Context)
Citation Context ...is isomorphic (from the E 2 -stage and up to k-duality) to the H∗(−, k)-version of the spectral sequence of [1], defined from Goodwillie’s approximations of the functor Σ ∞ Map(S n , X)+. We prove in =-=[19]-=- that En-operads are (up to operadic suspension) self-dual in the sense of Koszul duality of operads. We deduce from this result another representation of the homology theory HEn ∗ (A) which relates t... |

1 |
duality of chain operads equivalent to little cubes
- Koszul
(Show Context)
Citation Context ...nce is isomorphic (from the E2-stage and up to k-duality) to the H∗(−, k)-version of the spectral sequence of [1], defined from Goodwillie’s approximations of the functor Σ∞ Map(Sn , X)+. We prove in =-=[19]-=- that En-operads are (up to operadic suspension) self-dual in the sense of Koszul duality of operads. We deduce from this result another representation of the homology theory H En ∗ (A) which relates ... |