A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this article is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt-Eccles operad. We also prove that differential graded algebras over the Barratt-Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt-Eccles operad which is equivalent to the classical little squares operad.Comment: 46 pages. Final versio

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Introduction Conventions Chapter 1. Composition products and operad structures 1.1. Introduction: monads and operads 1.2. Composition products and operad structures 1.3. The composition product of symmetric modules Chapter 2. Chain complexes of modules over an operad 2.1. Summary 2.2. Digression: model categories of modules over an operad 2.3. On composite symmetric modules in the differential graded framework 45 2.4. The spectral sequence of a quasi-free module 2.5. Proof of comparison theorems Chapter 3. The reduced bar construction 3.1. Summary: quasi-free operads and reduced bar constructions 57 3.2. Digression: the model category of operads 3.3. The language of trees 3.4. Trees and free operads 3.5. Trees and reduced bar constructions 3.6. Comparison of quasi-free operads: proof of theorem 3.2.1 Chapter 4. Bar constructions with coefficients 4.1. Summary 4.2. Composite symmetric modules and trees with levels 4.3. The simplicial bar construction 4.4. The differential graded bar construction 4.5. The levelization morphism 4.6. Proofs 4.7. Quasi-free resolutions of operads and bar constructions Chapter 5. Koszul duality for operads iii iv CONTENTS 5.1. Weight graded operads 5.2. Koszul operads 5.3. Koszul complexes and characterization of Koszul operads Chapter 6. Epilog: partition posets Bibliography Glossary Notation PROLOGOne can deduce from results of A. Björner that the (reduced) homology of partition posets vanishes in degree * = r − 1 (cf. [14]). Then, in [5], H. Barcelo defines an isomorphism of representations Hr−1 ( K(r), K) ≃ L(r) ∨ ⊗ sgn r , based on the Lyndon basis of the Lie operad (cf. M. Lothaire [50], C. Reutenauer [70]). This result is improved by P. Hanlon and M. Wachs in [35]. Namely, these authors define a natural morphism (which does not involve the choice of a basis of the Lie operad) from the dual of the Lie operad to the chain complex of the partition posetThis morphism fixes a representative of the homology class associated to a given element of the Lie operad. In addition, P. Hanlon and M. Wachs generalize the theorem above and give a relationship between partition posets and structures of Lie algebras with kary brackets.On the other hand, a topological proof of the theorem above, based on calculations of F. Cohen (cf. [20]), is given by G. Arone and M. Kankaanrinta in [2]. In connection with this result, we should mention that an article of G. Arone and M. Mahowald (cf. [3]) sheds light on the importance of partition posets in homotopy theory. Namely, these authors prove that the Goodwillie tower of the identity functor on topological spaces is precisely determined by partition posets. * (S(C)(V )) vanishes for general reasons while the Harrison homology H Harr * (S(C)(V )) does not (cf. M. Barr [6], D. Harrison [36], S. Whitehouse [85]). Consequently, the comparison morphism * ∈N V * equipped with a differential δ : V * → V * −1 which decreases degrees by 1. A dg-module is equivalent to a chain complex

Contents: The algebraic theory and its topological background-The applications of (rational) homotopy theory methods Identifiers: LCCN 2016032055| ISBN 9781470434816 (alk. paper) | ISBN 9781470434823 (alk. paper) Subjects: LCSH: Homotopy theory. | Operads. | Grothendieck groups. | Teichmüller spaces. | AMS: Algebraic topology-Homotopy theory-Loop space machines, operads. msc | Category theory; homological algebra-Homological algebra-Homotopical algebra. msc | Algebraic topology-Homotopy theory-Homotopy equivalences. msc | Algebraic topology-Homotopy theory-Rational homotopy theory. msc | Manifolds and cell complexes-Homology and homotopy of topological groups and related structures-Hopf algebras. msc | Group theory and generalizations-Permutation groups-Infinite automorphism groups. msc | Group theory and generalizations-Special aspects of infinite or finite groups-Braid groups; Artin groups. msc Classification: LCC QA612.7 .F74 2017 | DDC 514/.24-dc23 LC record available at https://lccn. loc.gov/2016032055 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

We prove that any category of props in a symmetric monoidal model category inherits a model structure. We devote an appendix, about half the size of the paper, to the proof of the model category axioms in a general setting. We need the general argument to address the case of props in topological spaces and dg-modules over an arbitrary ring, but we give a less technical proof which applies to the category of props in simplicial sets, simplicial modules, and dg-modules over a ring of characteristic 0. We apply the model structure of props to the homotopical study of algebras over a prop. Our goal is to prove that an object 𝑋 homotopy equivalent to an algebra 𝐴 over a cofibrant prop P inherits a P-algebra structure so that 𝑋 defines a model of 𝐴 in the homotopy category of P-algebras. In the differential graded context, this result leads to a generalization of Kadeishvili's minimal model of 𝐴∞-algebras.