Complete $L_\infty$-algebras and their homotopy theory

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 We provide an explicit proof of an unpublished result of E.\ Getzler which states that the category $\hat{\mathsf{Lie}}_\infty$ of such $L_\infty$-algebras and filtration-preserving $\infty$-morphisms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.