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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-categories.