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

TL;DR

This paper develops a new combinatorial and $alculus-based framework connecting differential graded cocommutative coalgebras and curved Lie algebras, extending classical duality to broader categories.

Contribution

It introduces a model structure for dg cocommutative coalgebras and an $alculus structure for curved Lie algebras, establishing an $alculus equivalence between them.

Findings

Established a model structure for dg cocommutative coalgebras.

Extended functors to curved Lie algebras and dg coalgebras.

Proved an $alculus equivalence of categories.

Abstract

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.