Sullivan minimal models of operad algebras

TL;DR

This paper establishes the existence of Sullivan minimal models for a broad class of operad algebras, extending classical minimal model theory to operads like Com, Ass, Lie, and Gerstenhaber in characteristic zero.

Contribution

It adapts Sullivan's step-by-step construction to operad algebras, providing minimal models for various operads including those with non-zero degree components.

Findings

Constructed Sullivan minimal models for operad algebras in characteristic zero.

Extended minimal model theory to operads like Gerstenhaber.

Included operads with non-zero degree components.

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. Other interesting operads, such as the operad Ger encoding Gerstenhaber algebras, also fit in our study.