Derived $A_{\infty}$–algebras in an operadic context

Abstract

Derived $A_{\infty }$–algebras were developed recently by Sagave. Their advantage over classical $A_{\infty }$–algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived $A_{\infty }$–algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad $d\mathcal{A}s$ encoding bidgas, ie bicomplexes with an associative multiplication. This generalises the established result describing the operad $A_{\infty }$ as a resolution of the operad $\mathcal{A}s$ encoding associative algebras. We further show that Sagave’s definition of morphisms agrees with the infinity-morphisms of $d{A}_{\infty }$–algebras arising from operadic machinery. We also study the operadic homology of derived $A_{\infty }$–algebras.