Derived Poincaré–Birkhoff–Witt theorems: with an appendix by Vladimir Dotsenko

We propose a new general formalism that allows us to study Poincaré–Birkhoff–Witt type phenomena for universal enveloping algebras in the differential graded context. Using it, we prove a homotopy invariant version of the classical Poincaré–Birkhoff–Witt theorem for universal envelopes of Lie algebras. In particular, our results imply that all the previously known constructions of universal envelopes of L ∞ -algebras (due to Baranovsky, Lada and Markl, and Moreno-Fernández) represent the same object of the homotopy category of differential graded associative algebras. We also extend Quillen’s classical quasi-isomorphism C ⟶ B U from differential graded Lie algebras to L ∞ -algebras; this confirms a conjecture of Moreno-Fernández.