Geodesic
Derived A∞–algebras in an operadic context
Livernet, Muriel1 ; Roitzheim, Constanze2 ; Whitehouse, Sarah3
1Université Paris 13, Sorbonne Paris Cité, CNRS, UMR 7539, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France
2School of Mathematics, Statistics and Actuarial Science, University of Kent, Cornwallis, Canterbury CT2 7NF, UK
3School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK
Algebraic and Geometric Topology, Tome 13 (2013) no. 1, pp. 409-440
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

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.

DOI : 10.2140/agt.2013.13.409
Classification : 16E45, 18D50, 18G55, 18G10
Keywords: operads, $A_{\infty}$–algebras, Koszul duality

Livernet, Muriel  1   ; Roitzheim, Constanze  2   ; Whitehouse, Sarah  3

1 Université Paris 13, Sorbonne Paris Cité, CNRS, UMR 7539, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France
2 School of Mathematics, Statistics and Actuarial Science, University of Kent, Cornwallis, Canterbury CT2 7NF, UK
3 School of Mathematics and Statistics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK 
@article{10_2140_agt_2013_13_409,
     author = {Livernet, Muriel and Roitzheim, Constanze and Whitehouse, Sarah},
     title = {Derived {A\ensuremath{\infty}{\textendash}algebras} in an operadic context},
     journal = {Algebraic and Geometric Topology},
     pages = {409--440},
     year = {2013},
     volume = {13},
     number = {1},
     doi = {10.2140/agt.2013.13.409},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2013.13.409/}
}
TY  - JOUR
AU  - Livernet, Muriel
AU  - Roitzheim, Constanze
AU  - Whitehouse, Sarah
TI  - Derived A∞–algebras in an operadic context
JO  - Algebraic and Geometric Topology
PY  - 2013
SP  - 409
EP  - 440
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2013.13.409/
DO  - 10.2140/agt.2013.13.409
ID  - 10_2140_agt_2013_13_409
ER  - 
%0 Journal Article
%A Livernet, Muriel
%A Roitzheim, Constanze
%A Whitehouse, Sarah
%T Derived A∞–algebras in an operadic context
%J Algebraic and Geometric Topology
%D 2013
%P 409-440
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2013.13.409/
%R 10.2140/agt.2013.13.409
%F 10_2140_agt_2013_13_409
Livernet, Muriel; Roitzheim, Constanze; Whitehouse, Sarah. Derived A∞–algebras in an operadic context. Algebraic and Geometric Topology, Tome 13 (2013) no. 1, pp. 409-440. doi: 10.2140/agt.2013.13.409

[1] B Fresse, Koszul duality of operads and homology of partition posets, from: "Homotopy theory : relations with algebraic geometry, group cohomology, and algebraic K–theory" (editors P Goerss, S Priddy), Contemp. Math. 346, Amer. Math. Soc. (2004) 115

[2] B Fresse, Operadic cobar constructions, cylinder objects and homotopy morphisms of algebras over operads, from: "Alpine perspectives on algebraic topology" (editors C Ausoni, K Hess, J Scherer), Contemp. Math. 504, Amer. Math. Soc. (2009) 125

[3] J E Harper, Homotopy theory of modules over operads and non-Σ operads in monoidal model categories, J. Pure Appl. Algebra 214 (2010) 1407

[4] T V Kadeišvili, On the theory of homology of fiber spaces, Uspekhi Mat. Nauk 35 (1980) 183

[5] M Livernet, Pre-Lie systems and obstruction to A∞–structures over a ring

[6] J L Loday, B Vallette, Algebraic operads, 346, Springer (2012)

[7] F Muro, Homotopy theory of nonsymmetric operads, Algebr. Geom. Topol. 11 (2011) 1541

[8] C Roitzheim, S Whitehouse, Uniqueness of A∞–structures and Hochschild cohomology, Algebr. Geom. Topol. 11 (2011) 107

[9] S Sagave, DG-algebras and derived A∞–algebras, J. Reine Angew. Math. 639 (2010) 73

Cité par Sources :