Conilpotent Koszul duality, as formulated by Positselski and Lefevre-Hasegawa, gives an equivalence (of model categories, or of ∞-categories) between augmented dg algebras and conilpotent dg-coalgebras. One should think of this as a non-commutative version of the Lurie-Pridham correspondence: indeed in characteristic zero, cocommutative conilpotent dg coalgebras are Koszul dual to dg Lie algebras, and this is precisely the correspondence between formal moduli problems and their tangent complexes. I’ll talk about a global analogue where the conilpotency assumption is removed; geometrically this corresponds to non-commutative formal moduli problems modelled on profinite completions, rather than pro-Artinian completions. Global Koszul duality is best expressed as a Quillen equivalence between curved dg algebras and curved dg coalgebras, and in both categories the weak equivalences are defined using an auxiliary object, the Maurer-Cartan dg category of a curved dg algebra.
This is joint work with Andrey Lazarev, and the talk is based on this arXiv paper.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
