Formality criteria for algebras over operads

TL;DR

This paper establishes criteria for the formality of differential graded algebras over operads, unifying previous approaches and introducing new operadic classes that serve as obstructions to formality.

Contribution

It generalizes existing formality criteria by developing operadic Kaledin classes and linking formality to spectral sequence degeneration.

Findings

Operadic Kaledin classes act as obstructions to formality.

Formality of an algebra is equivalent to spectral sequence degeneration at E2.

Provides a unified framework for formality criteria over operads.

Abstract

We study some formality criteria for differential graded algebras over differential graded operads. This unifies and generalizes other known approaches like the ones by Manetti and Kaledin. In particular, we construct general operadic Kaledin classes and show that they provide obstructions to formality. Moreover, we show that an algebra A is formal if and only if its operadic cohomology spectral sequence degenerates at $E_2$.