Complete L∞-algebras and their homotopy theory

Abstract

We analyze a model for the homotopy theory of complete filtered L∞-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 We provide an explicit proof of an unpublished result of E. Getzler which states that the category Lieˆ∞ of such L∞-algebras and filtration-preserving ∞-morphisms admits the structure of a category of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∞-quasi-isomorphism in Lieˆ∞ has a filtration preserving homotopy inverse. Also, we show that the simplicial Maurer–Cartan functor, which assigns a Kan simplicial set to each L∞-algebra in Lieˆ∞, 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 ∞-morphism. The obstruction classes reside in the associated graded mapping cone of the corresponding tangent map.