Derived A-infinity algebras in an operadic context

TL;DR

This paper explores derived A-infinity algebras within an operadic framework, showing their relation to resolutions of operads and establishing connections with classical A-infinity structures, enhancing understanding of their algebraic properties.

Contribution

It demonstrates how derived A-infinity algebras can be viewed as operad algebras, generalizes known operadic resolutions, and aligns Sagave's morphisms with operadic infinity-morphisms.

Findings

Derived A-infinity algebras are operad algebras via resolutions.

The operad for derived A-infinity algebras generalizes classical A-infinity operad.

Sagave's morphisms coincide with operadic infinity-morphisms.

Abstract

Derived A-infinity algebras were developed recently by Sagave. Their advantage over classical A-infinity algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived A-infinity algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad dAs encoding bidgas. This generalises the established result describing the operad A-infinity as a resolution of the operad As encoding associative algebras. We further show Sagave's definition of morphisms agrees with the infinity-morphisms of dA-infinity algebras arising from operadic machinery. We also study the operadic homology of derived A-infinity algebras.