Complete \(L_\infty \)-algebras and their homotopy theory
In previous joint work [J. Algebra 430, 260--302 (2015; Zbl 1327.17019)] with \textit{V. Dolgushev}, the author generalized a result of E. Getzler [Zbl 1246.17025], establishing a connection between the homotopy theory of complete \(L_{\infty}\)-algebras and Kan complexes, which is a generalization of the classical theorem of \textit{W. M. Goldman} and \textit{J. J. Millson} [Publ. Math., Inst. Hautes Étud. Sci. 67, 43--96 (1988; Zbl 0678.53059), Theorem 4.8] and which has turned out to be useful in a variety of applications beyond deformation theory, including the rational homotopy theory of mapping spaces [\textit{A. Berglund}, Homology Homotopy Appl. 17, No. 2, 343--369 (2015; Zbl 1347.55010); \textit{B. Fresse} and \textit{T. Willwacher}, ``Mapping spaces for DG Hopf cooperads and homotopy automorphisms of the rationalization of $E_n$-operads'', Preprint, \url{arXiv:2003.02939}] and operadic homotopical algebras [\textit{V. A. Dolgushev} et al., Adv. Math. 274, 562--605 (2015; Zbl 1375.18053); \textit{D. Robert-Nicoud}, Algebr. Geom. Topol. 19, No. 3, 1453--1476 (2019; Zbl 1475.17032)]. This paper further develops and strengthens the above result. The author first gives a finer-grained description of the homotopy theory within the category \(\widehat{\mathsf{Lie}}_{\infty}\), where \ is the category of complete filtered \(L_{\infty}\)-algebras and filtration-compatible weak \(L_{\infty}\)-morphisms. He then extends the aforementioned ``\(L_{\infty}\) Goldman-Millson Theorem'' by establishing the compatibility of this additional homotopical structure with the Kan-Quillen model structure for simpllcial sets. Finally, he uses this machinery to build an obstruction theory for the general problem of lifting Maurer-Cartan elements through an arbitrary \(\infty\)-morphism.
scientific article; zbMATH DE number 7701450
scientific article