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On the infinity category of homotopy Leibniz algebras

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David Khudaverdyan, Norbert Poncin, Jian Qiu

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

2010 MSC:
18D99, 55P99, 55U10, 17A32

*Theory and Applications of Categories,*
Vol. 29, 2014,
No. 12, pp 332-370.

Published 2014-06-19.

http://www.tac.mta.ca/tac/volumes/29/12/29-12.pdf

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