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An interpretation of Enhomology as functor homology
"... Abstract. We prove that Enhomology of nonunital commutative algebras can be described as functor homology when one considers functors from a certain category of planar trees with n levels. For different n these homology theories are connected by natural maps, ranging from Hochschild homology and i ..."
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Abstract. We prove that Enhomology of nonunital commutative algebras can be described as functor homology when one considers functors from a certain category of planar trees with n levels. For different n these homology theories are connected by natural maps, ranging from Hochschild homology and its higher order versions to Gamma homology. 1.
AN E∞EXTENSION OF THE ASSOCIAHEDRA AND THE TAMARKIN CELL MYSTERY
, 2009
"... Abstract. In this note based on the author’s communication with M. Batanin, we study a cofibrant E∞operad generated by the FoxNeuwirth cells of the configuration space of points in the Euclidean space. We show that, below the ‘critical dimensions ’ in which ‘bad cells ’ exist, this operad is model ..."
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Abstract. In this note based on the author’s communication with M. Batanin, we study a cofibrant E∞operad generated by the FoxNeuwirth cells of the configuration space of points in the Euclidean space. We show that, below the ‘critical dimensions ’ in which ‘bad cells ’ exist, this operad is modeled by the geometry of the FultonMacPherson compactification of this configuration space. We analyze the Tamarkin bad cell and calculate the differential of the corresponding generator. We also describe a simpler, fourdimensional bad cell. We finish the paper by proving an auxiliary result giving a characterization, over integers, of free Lie algebras. Contents
ITERATED BAR COMPLEXES AND THE POSET OF PRUNED TREES ADDENDUM TO THE PAPER: “ITERATED BAR COMPLEXES OF EINFINITY ALGEBRAS AND HOMOLOGY THEORIES”
, 2010
"... Abstract. The purpose of these notes is to explain the relationship between Batanin’s categories of pruned trees and iterated bar complexes. This article is an appendix of the article [4]. Our purpose is to explain the relationship between Batanin’s categories of pruned trees (see [1, 2]) and iterat ..."
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Abstract. The purpose of these notes is to explain the relationship between Batanin’s categories of pruned trees and iterated bar complexes. This article is an appendix of the article [4]. Our purpose is to explain the relationship between Batanin’s categories of pruned trees (see [1, 2]) and iterated bar complexes and to revisit some constructions [4] in this formalism. The reader can use this appendix as an informal introduction to the constructions of [4]. These notes is the appendix part of a preliminary version of [4], extracted without changes from this article except that we have removed the appendix mark from paragraph numberings. Thus the reader can easily retrieve references to former versions of [4] in this manuscript. 1. Level trees and sequences of nondecreasing surjections. Our first aim is to make explicit the expansion of iterated tensor coalgebras (T c Σ) n (M), for a connected Σ∗module
THE TAMARKIN CELL MYSTERY
"... Abstract. In this note based on the author’s communication with M. Batanin, we study a cofibrant E∞operad generated by the FoxNeuwirth cells of the configuration space of points in the Euclidean space. We show that, below the ‘critical dimensions ’ in which ‘bad cells ’ exist, this operad is model ..."
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Abstract. In this note based on the author’s communication with M. Batanin, we study a cofibrant E∞operad generated by the FoxNeuwirth cells of the configuration space of points in the Euclidean space. We show that, below the ‘critical dimensions ’ in which ‘bad cells ’ exist, this operad is modeled by the geometry of the FultonMacPherson compactification of this configuration space. We analyze the Tamarkin bad cell and calculate the differential of the corresponding generator. We also describe a simpler, fourdimensional bad cell. We finish the paper by proving an auxiliary result giving a characterization, over integers, of free Lie algebras. Contents
The Bar Complex of an Einfinity Algebra
, 2008
"... The standard reduced bar complex B(A) of a differential graded algebra A inherits a natural commutative algebra structure if A is a commutative algebra. We address an extension of this construction in the context of Einfinity algebras. We prove that the bar complex of any Einfinity algebra can be ..."
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The standard reduced bar complex B(A) of a differential graded algebra A inherits a natural commutative algebra structure if A is a commutative algebra. We address an extension of this construction in the context of Einfinity algebras. We prove that the bar complex of any Einfinity algebra can be equipped with the structure of an Einfinity algebra so that the bar construction defines a functor from Einfinity algebras to Einfinity algebras. We prove the homotopy uniqueness of such natural Einfinity structures on the bar construction. We apply our construction to cochain complexes of topological spaces, which are instances of Einfinity algebras. We prove that the nth iterated bar complexes of the cochain algebra of a space X is equivalent to the cochain complex of the nfold iterated loop space of X, under reasonable connectedness, completeness and finiteness assumptions on X.