Sullivan minimal models of operad algebras

Abstract

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.

Up to homotopy algebras with strict units

“…1.3. As with our paper [CR19], a comparison with the minimal models of operads obtained thanks to the curved Koszul duality [Bur18], [HM12] might be in order. Of course, since both share the property of being minimal, they must give isomorphic models when applied to the same operads.…”

“…Similarly to the setting of commutative algebras, there is a notion of homotopy between morphisms of operads, defined via a functorial path (see Section 3.10 of [MSS02], cf. [CR19]), based on the following remark.…”

Mixed Hodge structures on cohomology jump ideals

“…This works equally well with AHC because at each step are added to the minimal model pieces of the MHS of H(A) : the essential lemma is [CG14, Lemma 3.12]. The theory of minimal models, without MHS but for many more algebras over operads, is written in [CR19]; this one works for L ∞ algebras as stated there. Combining both gives minimal models for AHC + L ∞ algebras.…”

On the group of homotopy classes of relative homotopy automorphisms

“…This section is inspired by [CR19], in which minimal models for algebras over algebraic operads are constructed. We adopt the general framework and generalize some of the theory to a relative dg Lie algebras.…”