Derived Poincaré-Birkhoff-Witt theorems. With an appendix by Vladimir Dotsenko (Q2687710)

The principal objective in this paper is to show, using the methods of [\textit{V. Dotsenko} and \textit{P. Tamaroff}, Int. Math. Res. Not. 2021, No. 16, 12670--12690 (2021; Zbl 07471395)] suitably extended to the dg setting, that the results of [\textit{V. Baranovsky}, Math. Res. Lett. 15, No. 5--6, 1073--1089 (2008; Zbl 1170.16018); \textit{J. M. Moreno Fernández}, Math. Z. 300, No. 3, 2147--2165 (2022; Zbl 1491.55012)] follow from a derived version of the classical Poincaré-Birkhoff-Witt theorem for Lie algebras. To accomplish the desideratum of [\textit{V. Baranovsky}, Math. Res. Lett. 15, No. 5--6, 1073--1089 (2008; Zbl 1170.16018), pp.2] asking for a more thorough understanding of the universal enveloping functor from the operadic point of view, the authors address, through the theory of operads, the Lada-Markl functor assigning to an \(A_{\infty}\)-algebra the \(L_{\infty}\)-algebra obtained by antisymmetrizing all the product operations. It is established that its left adjoint abides by a derived Poincaré-Birkhoff-Witt theorem, so that the three proposed models for universal envelopes of \(L_{\infty}\)-algebras [\textit{V. Baranovsky}, Math. Res. Lett. 15, No. 5--6, 1073--1089 (2008; Zbl 1170.16018); \textit{J. M. Moreno Fernández}, Math. Z. 300, No. 3, 2147--2165 (2022; Zbl 1491.55012); \textit{T. Lada} and \textit{M. Markl}, Commun. Algebra 23, No. 6, 2147--2161 (1995; Zbl 0999.17019)] are, up to homotopy, one and the same. Quillen's classical quasi-isomorphism \[ \mathcal{C}\rightarrow BU \] from differential graded Lie algebras to \(L_{\infty}\)-algebras is extended, which confirms a conjecture of \textit{J. M. Moreno Fernández} [Math. Z. 300, No. 3, 2147--2165 (2022; Zbl 1491.55012)].