2013/08/12 by David Khudaverdyan, Khudaverdyan, David, Norbert Poncin +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1308.2583
openalex publication_date 2013/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \citeGet09. 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 \citeBC04, as well as the corresponding composition rule \citeSS07. We also answer a question of \citeBS07 about composition of ∞-homotopies of ∞-algebras.