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 $\mathcal{C\mkern-1mu om}$, $\mathcal{A\mkern-1mu ss}$, and $\mathcal{L\mkern-1mu ie}$, as well as over their minimal models $\mathcal{C\mkern-1mu om}_\infty$, $\mathcal{A\mkern-1mu ss}_\infty$, and $\mathcal{L\mkern-1mu ie}_\infty$. Other interesting operads, such as the operad $\mathcal{G\mkern-1mu er}$ encoding Gerstenhaber algebras, also fit in our study.