#### 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
##### Publication
Received: 8 June 2012
Revised: 10 September 2012
Accepted: 4 October 2012
Published: 6 March 2013
##### Authors
 Muriel Livernet Université Paris 13 Sorbonne Paris Cité CNRS, UMR 7539 99 avenue Jean-Baptiste Clément 93430 Villetaneuse France http://www.math.univ-paris13.fr/~livernet/ Constanze Roitzheim School of Mathematics, Statistics and Actuarial Science University of Kent Cornwallis Canterbury CT2 7NF UK http://www.kent.ac.uk/smsas/personal/csrr/ Sarah Whitehouse School of Mathematics and Statistics University of Sheffield Hicks Building Sheffield S3 7RH UK http://www.sarah-whitehouse.staff.shef.ac.uk/