On the infinity category of homotopy Leibniz algebras

TL;DR

This paper explores the structure of $ abla$-categories of homotopy Leibniz algebras, establishing their properties, and connecting them with known concepts of homotopy and composition in higher algebraic structures.

Contribution

It introduces a framework for $ abla$-categories of $ abla$-homotopies, relates them to quasi-categories, and clarifies composition rules for $ abla$-homotopies in $ abla$-algebras.

Findings

The nerve of a complete Lie $ abla$-algebra forms a Kan complex.

Truncated $ abla$-algebras project to strict 2-categories.

Concrete application to 2-term $ abla$-algebras recovers known homotopy concepts.

Abstract

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 \cite{Get09}. 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 \cite{BC04}, as well as the corresponding composition rule \cite{SS07}. We also answer a question of \cite{BS07} about composition of $\infty$-homotopies of $\infty$-algebras.