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Derived
$A_{\infty}$–algebras in an operadic context
Muriel Livernet, Constanze Roitzheim and Sarah Whitehouse |
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Algebraic & Geometric Topology 13 (2013) 409–440
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Abstract |
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Derived –algebras were developed recently by Sagave. Their advantage over classical –algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived –algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad encoding bidgas, ie bicomplexes with an associative multiplication. This generalises the established result describing the operad as a resolution of the operad encoding associative algebras. We further show that Sagave’s definition of morphisms agrees with the infinity-morphisms of –algebras arising from operadic machinery. We also study the operadic homology of derived –algebras. |
Keywords
operads, $A_{\infty}$–algebras, Koszul duality
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Mathematical Subject Classification 2010
Primary: 16E45, 18D50
Secondary: 18G55, 18G10
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References |
Publication
Received: 8 June 2012
Revised: 10 September 2012
Accepted: 4 October 2012
Published: 6 March 2013
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Authors |
| Muriel Livernet | |
| Université Paris 13 Sorbonne Paris Cité CNRS, UMR 7539 99 avenue Jean-Baptiste Clément 93430 Villetaneuse France |
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| http://www.math.univ-paris13.fr/~livernet/ | |
| Constanze Roitzheim | |
| School of Mathematics, Statistics
and Actuarial Science University of Kent Cornwallis Canterbury CT2 7NF UK |
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| http://www.kent.ac.uk/smsas/personal/csrr/ | |
| Sarah Whitehouse | |
| School of Mathematics and
Statistics University of Sheffield Hicks Building Sheffield S3 7RH UK |
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| http://www.sarah-whitehouse.staff.shef.ac.uk/ | |
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