ON THE INFINITY CATEGORY OF HOMOTOPY LEIBNIZ ALGEBRAS

Abstract. We discuss various concepts of ∞-homotopies, as well as the relations between them (focussing on the Leibniz type). In particular ∞-n-homotopies appear as the n-simplices of the nerve of a complete Lie ∞-algebra. In the nilpotent case, this nerve is known to be a Kan complex [Get09]. We argue that there is a quasi-category of ∞-algebras and show that for truncated ∞-algebras, i.e. categorified algebras, this ∞-categorical structure projects to a strict 2-categorical one. The paper contains a shortcut to (∞, 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 ∞-algebra case, thus recovering the concept of homotopy of [BC04], as well as the corresponding composition rule [SS07]. We also answer a question of [Sho08] about composition of ∞-homotopies of ∞-algebras. 1