2020/08/04 by Christopher L. Rogers, Rogers, Christopher L.
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2008.01706
openalex publication_date 2020/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 We provide an explicit proof of an unpublished result of E. Getzler which states that the category Lie_∞ of such L_∞-algebras and filtration-preserving ∞-morphisms admits the structure of a category of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 Lie_∞ 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 Lie_∞, 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.