#### Volume 13, issue 1 (2013)

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

### Muriel Livernet, Constanze Roitzheim and Sarah Whitehouse

Algebraic & Geometric Topology 13 (2013) 409–440
##### 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 $d\mathsc{A}s$ encoding bidgas, ie bicomplexes with an associative multiplication. This generalises the established result describing the operad ${A}_{\infty }$ as a resolution of the operad $\mathsc{A}s$ 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.

##### Keywords
operads, $A_{\infty}$–algebras, Koszul duality
##### Mathematical Subject Classification 2010
Primary: 16E45, 18D50
Secondary: 18G55, 18G10