Volume 10, issue 3 (2010)

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$\Gamma$–homology of algebras over an operad

Eric Hoffbeck

Algebraic & Geometric Topology 10 (2010) 1781–1806
Abstract

The purpose of this paper is to study generalizations of Gamma-homology in the context of operads. Good homology theories are associated to operads under appropriate cofibrancy hypotheses, but this requirement is not satisfied by usual operads outside the characteristic zero context. In that case, the idea is to pick a cofibrant replacement $\mathsf{Q}$ of the given operad $\mathsf{P}$. We can apply to $\mathsf{P}$–algebras the homology theory associated to $\mathsf{Q}$ in order to define a suitable homology theory on the category of $\mathsf{P}$–algebras. We make explicit a small complex to compute this homology when the operad $\mathsf{P}$ is binary and Koszul. In the case of the commutative operad $\mathsf{P}=\mathsf{Com}$, we retrieve the complex introduced by Robinson for the Gamma-homology of commutative algebras.

Keywords
algebras, operad, homology theory, gamma-homology
Mathematical Subject Classification 2000
Primary: 16E40
Secondary: 18D50, 18G55, 18G60
Publication
Received: 1 February 2010
Accepted: 30 June 2010
Published: 30 August 2010
Authors
 Eric Hoffbeck Laboratoire Paul Painlevé Université Lille 1 Cité Scientifique 59655 Villeneuve d’Ascq Cedex France http://math.univ-lille1.fr/~hoffbeck