Abstract. Let M g,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 (M g,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-Griffiths-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.
MINIMAL RESOLUTIONS AND OTHER MINIMAL MODELS AGUSTÍ RoIGIn many situations, minimal models are used as representatives of homotopy types. In this paper we state this fact as an equivalence of categories . This equivalence follows from an axiomatic definition of minimal objects. We see that this definition includes examples such as minimal resolutions of Eilenberg-NakayamaTate, minimal fiber spaces of Kan and A-minimal A-extensions of Halperin . For the first one, this is done by generalizing the construction of minimal resolutions of modules to complexes. The others follow by a caracterization of minimal objects in bifibred categories .
We 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 Com∞, Ass∞ and Lie∞. Other interesting operads, such as the operad Ger encoding Gerstenhaber algebras, also fit in our study.2010 Mathematics Subject Classification. 18D50, 55P62.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.