Complete \(L_\infty\)-algebras and their homotopy theory

• Published in: arXiv.org 2023-05
• Main Author: Rogers, Christopher L
• Format: Article
• Language: English
• Subjects: Algebra; Deformation; Filtration; Homotopy theory; Isomorphism; Uniqueness theorems
• ISSN: 2331-8422

We analyze a model for the homotopy theory of complete filtered \(L_\infty\)-algebras intended for applications in algebraic and algebro-geometric deformation theory. We provide an explicit proof of an unpublished result of E.\ Getzler which states that the category \(\hat{\mathsf{Lie}}_\infty\) of such \(L_\infty\)-algebras and filtration-preserving \(\infty\)-morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. Novel applications of our approach include explicit models for homotopy pullbacks, and an analog of Whitehead's Theorem: under some mild conditions, every filtered \(L_\infty\)-quasi-isomorphism in \(\hat{\mathsf{Lie}}_\infty\) has a filtration preserving homotopy inverse. Also, we show that the simplicial Maurer--Cartan functor, which assigns a Kan simplicial set to each \(L_\infty\)-algebra in \(\hat{\mathsf{Lie}}_\infty\), is an exact functor between the respective CFOs. Finally, we provide an obstruction theory for the general problem of lifting a Maurer-Cartan element through an \(\infty\)-morphism. The obstruction classes reside in the associated graded mapping cone of the corresponding tangent map.