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.
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