ON THE INFINITY CATEGORY OF HOMOTOPY LEIBNIZ ALGEBRAS

We discuss various concepts of infinity-homotopies, as well as the relations between them (focussing on the Leibniz type). In particular infinity-n-homotopies appear as the n-simplices of the nerve of a complete Lie infinity-algebra. In the nilpotent case, this nerve is known to be a Kan complex [Get09]. We argue that there is a quasi-category of infinity-algebras and show that for truncated infinity-algebras, i.e. categorified algebras, this infinity-categorical structure projects to a strict 2-categorical one. The paper contains a shortcut to (infinity, 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 infinity-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 infinity-homotopies of infinity-algebras.