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.
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},
year = {2014},
volume = {29},
language = {en},
url = {http://geodesic.mathdoc.fr/item/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/