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

Derived A–algebras were developed recently by Sagave. Their advantage over classical A–algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived A–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, ie bicomplexes with an associative multiplication. This generalises the established result describing the operad A as a resolution of the operad As encoding associative algebras. We further show that Sagave’s definition of morphisms agrees with the infinity-morphisms of dA–algebras arising from operadic machinery. We also study the operadic homology of derived A–algebras.

operads, $A_{\infty}$–algebras, Koszul duality
Mathematical Subject Classification 2010
Primary: 16E45, 18D50
Secondary: 18G55, 18G10
Received: 8 June 2012
Revised: 10 September 2012
Accepted: 4 October 2012
Published: 6 March 2013
Muriel Livernet
Université Paris 13
Sorbonne Paris Cité
CNRS, UMR 7539
99 avenue Jean-Baptiste Clément
93430 Villetaneuse
Constanze Roitzheim
School of Mathematics, Statistics and Actuarial Science
University of Kent
Canterbury CT2 7NF
Sarah Whitehouse
School of Mathematics and Statistics
University of Sheffield
Hicks Building
Sheffield S3 7RH