Complete L∞-algebras and their homotopy theory

We analyze a model for the homotopy theory of complete filtered L-infinity-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 We provide an explicit proof of an unpublished result of E. Getzler which states that the category admits the structure of a category of fibrant objects (CFO) for a homotopy theory.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-infinity-quasi-isomorphism in (Lie) over cap (infinity) has a filtration preserving homotopy inverse. Also, we show that the simplicial Maurer-Cartan functor, which assigns a Kan simplicial set to each L-infinity-algebra in (Lie) over cap (infinity), 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 infinity-morphism. The obstruction classes reside in the associated graded mapping cone of the corresponding tangent map. (c) 2023 Elsevier B.V. All rights reserved.