Geodesic
On the infinity category of homotopy Leibniz algebras
Theory and applications of categories, Tome 29 (2014), pp. 332-370.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

We discuss various concepts of $\infty$-homotopies, as well as the relations between them (focussing on the Leibniz type). In particular $\infty$-$n$-homotopies appear as the $n$-simplices of the nerve of a complete Lie ${\infty}$-algebra. In the nilpotent case, this nerve is known to be a Kan complex. We argue that there is a quasi-category of $\infty$-algebras and show that for truncated $\infty$-algebras, i.e. categorified algebras, this $\infty$-categorical structure projects to a strict 2-categorical one. The paper contains a shortcut to $(\infty,1)$-categories, as well as a review of Getzler's proof of the Kan property. We make the latter concrete by applying it to the 2-term $\infty$-algebra case, thus recovering the concept of homotopy of Baez and Crans, as well as the corresponding composition rule \cite{SS07}. We also answer a question of Shoikhet about composition of $\infty$-homotopies of $\infty$-algebras.
Publié le :
Classification : 18D99, 55P99, 55U10, 17A32
Keywords: Homotopy algebra, categorified algebra, higher category, quasi-category, Kan complex, Maurer-Cartan equation, composition of homotopies, Leibniz algebra 
@article{TAC_2014_29_a11,
     author = {David Khudaverdyan and Norbert Poncin and Jian Qiu},
     title = {On the infinity category of homotopy {Leibniz} algebras},
     journal = {Theory and applications of categories},
     pages = {332--370},
     publisher = {mathdoc},
     volume = {29},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2014_29_a11/}
}
TY  - JOUR
AU  - David Khudaverdyan
AU  - Norbert Poncin
AU  - Jian Qiu
TI  - On the infinity category of homotopy Leibniz algebras
JO  - Theory and applications of categories
PY  - 2014
SP  - 332
EP  - 370
VL  - 29
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TAC_2014_29_a11/
LA  - en
ID  - TAC_2014_29_a11
ER  - 
%0 Journal Article
%A David Khudaverdyan
%A Norbert Poncin
%A Jian Qiu
%T On the infinity category of homotopy Leibniz algebras
%J Theory and applications of categories
%D 2014
%P 332-370
%V 29
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TAC_2014_29_a11/
%G en
%F TAC_2014_29_a11
David Khudaverdyan; Norbert Poncin; Jian Qiu. On the infinity category of homotopy Leibniz algebras. Theory and applications of categories, Tome 29 (2014), pp. 332-370. http://geodesic.mathdoc.fr/item/TAC_2014_29_a11/