Sullivan minimal models of operad algebras

preprintWe prove the existence of Sullivan minimal models of operad algebras, for a quite wide family of operads in the category of complexes of vector spaces over a field of characteristic zero. Our construction is an adaptation of Sullivan’s original step by step construction to the setting of operad algebras. The family of operads that we consider includes all operads concentrated in degree 0 as well as their minimal models. In particular, this gives Sullivan minimal models for algebras over Com, Ass and Lie, as well as over their minimal models Com8, Ass8 and Lie8. Other interesting operads, such as the operad Ger encoding Gerstenhaber algebras, also fit in our study.Preprin

Up-to-homotopy algebras with strict units

We prove the existence of minimal models à la Sullivan for operads with non trivial arity zero. So up-to-homotopy algebras with strict units are just operad algebras over these minimal models. As an application we give another proof of the formality of the unitary n -little disks operad over the rationals.Preprin

Godement resolution and operad sheaf homotopy theory

We show how to induce products in sheaf cohomology for a wide variety of coefficients: sheaves of dg commutative and Lie algebras, symmetric Omega-spectra, filtered dg algebras, operads and operad algebras.Peer ReviewedPostprint (author's final draft

Moduli spaces and formal operads

Let Mg,l be the moduli space of stable algebraic curves of genus g with l marked points. With the operations which relate the different moduli spaces identifying marked points, the family (Mg,l)g,l is a modular operad of projective smooth Deligne-Mumford stacks, M. In this paper we prove that the modular operad of singular chains C?(M;Q) is formal; so it is weakly equivalent to the modular operad of its homology H?(M;Q). As a consequence, the “up to homotopy” algebras of these two operads are the same. To obtain this result we prove a formality theorem for operads analogous to Deligne-Grifiths-Morgan-Sullivan formality theorem, the existence of minimal models of modular operads, and a characterization of formality for operads which shows that formality is independent of the ground field

Monoidal functors, acyclic models and chain operads

We prove that for a topological operad P the operad of oriented cubical chains, Cord ¤ (P), and the operad of singular chains, S¤(P), are weakly equivalent. As a consequence, Cord ¤ (P;Q) is formal if and only if S¤(P;Q) is formal, thus linking together some formality results spread in the literature. The proof is based on an acyclic models theorem for monoidal functors. We give di®erent variants of the acyclic models theorem and apply the contravariant case to study the cohomology theories for simplicial sets de¯ned by R-simplicial di®erential graded algebras

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