Gamma-homology of algebras over an operad

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 Q of the given operad P. We can apply to P-algebras the homology theory associated to Q in order to define a suitable homology theory on the category of P-algebras. We make explicit a small complex to compute this homology when the operad P is binary and Koszul. In the case of the commutative operad P=Com, we retrieve the complex introduced by Robinson for the Gamma-homology of commutative algebras.Comment: 24 pages, correction in the definition of c_{i,j}, typos correcte

Leibniz homology of Lie algebras as functor homology

We prove that Leibniz homology of Lie algebras can be described as functor homology in the category of linear functors from a category associated to the Lie operad.Comment: 26 page

Operads with compatible CL-shellable partition posets admit a Poincar\'e-Birkhoff-Witt basis

In 2007, Vallette built a bridge across posets and operads by proving that an operad is Koszul if and only if the associated partition posets are Cohen-Macaulay. Both notions of being Koszul and being Cohen-Macaulay admit different refinements: our goal here is to link two of these refinements. We more precisely prove that any (basic-set) operad whose associated posets admit isomorphism-compatible CL-shellings admits a Poincar\'e-Birkhoff-Witt basis. Furthermore, we give counter-examples to the converse

Obstruction theory for algebras over an operad

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Shuffles of trees

We discuss a notion of shuffle for trees which extends the usual notion of a shuffle for two natural numbers. We give several equivalent descriptions, and prove some algebraic and combinatorial properties. In addition, we characterize shuffles in terms of open sets in a topological space associated to a pair of trees. Our notion of shuffle is motivated by the theory of operads and occurs in the theory of dendroidal sets, but our presentation is independent and entirely selfcontained

Homology of infinity-operads

25 pagesIn a first part of this paper, we introduce a homology theory for infinity-operads and for dendroidal spaces which extends the usual homology of differential graded operads defined in terms of the bar construction, and we prove some of its basic properties. In a second part, we define general bar and cobar constructions. These constructions send infinity-operads to infinity-cooperads and vice versa, and define an adjoint bar-cobar (or "Koszul") duality. Somewhat surprisingly, this duality is shown to hold much more generally between arbitrary presheaves and copresheaves on the category of trees defining infinity-operads. We emphasize that our methods are completely elementary and explicit