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

Christopher L. Rogers

Published 2020-08-04Version 1

We analyze a model for the homotopy theory of complete filtered $\mathbb{Z}$-graded $L_\infty$-algebras, which lends itself well to computations in deformation theory and homotopical algebra. We first give an explicit proof of an unpublished result of E. Getzler which states that the category ${\widehat{\mathsf{Lie}}\lbrack 1\rbrack_\infty}$ of such $L_\infty$-algebras and filtration-preserving $\infty$-morphisms admits the structure of a category of fibrant objects (CFO) for a homotopy theory. As a novel application, we use this result to show that, under some mild conditions, every $L_\infty$-quasi-isomorphism between $L_\infty$-algebras in ${\widehat{\mathsf{Lie}}\lbrack 1\rbrack_\infty}$ has a filtration preserving homotopy inverse. Finally, building on previous joint work with V. Dolgushev, we prove that the simplicial Maurer--Cartan functor, which assigns a Kan simplicial set to each complete filtered $L_\infty$-algebra, is an exact functor between the respective categories of fibrant objects. We interpret this as an optimal homotopy-theoretic generalization of the classical Goldman--Millson theorem from deformation theory. One immediate application is the ``$\infty$-categorical'' uniqueness theorem for homotopy transferred structures previously sketched by the author in arXiv:1612.07868.