Cocommutative coalgebras: homotopy theory and Koszul duality

TL;DR

This paper develops a model category framework for differential graded cocommutative coalgebras and establishes a Koszul duality extending to a Quillen equivalence with curved Lie algebras.

Contribution

It extends Hinich's construction to all dg cocommutative coalgebras and generalizes Koszul duality to a Quillen equivalence involving curved Lie algebras.

Findings

Established a closed model category structure for dg cocommutative coalgebras.

Extended Koszul duality to a Quillen equivalence between coalgebras and curved Lie algebras.

Demonstrated the duality over an algebraically closed field of characteristic zero.

Abstract

We extend a construction of Hinich to obtain a closed model category structure on all differential graded cocommutative coalgebras over an algebraically closed field of characteristic zero. We further show that the Koszul duality between commutative and Lie algebras extends to a Quillen equivalence between cocommutative coalgebras and formal coproducts of curved Lie algebras.