Global Koszul Duality: Differential Graded Cocommutative Coalgebras and Curved Lie Algebras

We give a combinatorial model structure to the category of, not necessarily conilpotent, differential graded (dg) cocommutative coalgebras and an $\infty$-category structure to the category of curved Lie algebras over an algebraically closed field of characteristic $0$. Further, we extend the Harrison and Chevally-Eilenberg functors between dg cocommutative conilpotent coalgebras and dg Lie algebras to these categories and show they form an equivalence of $\infty$-categories.

💡 Research Summary

The paper establishes a global Koszul duality between differential graded (dg) cocommutative coalgebras (without any conilpotency or augmentation hypotheses) and curved Lie algebras over an algebraically closed field of characteristic zero. The authors first recall the classical Harrison–Chevalley‑Eilenberg adjunction that links coaugmented conilpotent dg coalgebras with ordinary dg Lie algebras. They then extend both functors to the broader settings: the Harrison functor remains unchanged, while the Chevalley‑Eilenberg construction is replaced by an “extended” version (denoted ˇCE) that uses the cofree cocommutative coalgebra rather than its conilpotent counterpart. This requires working with pseudocompact commutative algebras and their completed symmetric tensor algebras ˇS(V), which are dual to the cofree cocommutative coalgebras.

A key technical step is to show that both functors represent the Maurer‑Cartan (MC) functor. For any dg Lie algebra g, MC(g) consists of degree‑1 elements satisfying d x + ½