$\Gamma$–homology of algebras over an operad

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.