Cocommutative coalgebras: homotopy theory and Koszul duality

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.

Koszul duality and homotopy theory of curved Lie algebras

“…However, the coalgebras were assumed to be conilpotent and this construction is not known in the completely general case. Further, Positselski worked with curved objects suggesting that in more general cases when discussing a Koszul duality one side of the Quillen equivalence should be a category consisting of curved objects; a hypothesis that is strengthened by results of [5] and this paper.…”

“…In Section 4 the category of marked curved Lie algebras and curved morphisms is introduced. This category is similar to the category of curved Lie algebras with strict morphisms discussed in [5]. On the other hand, the morphisms are quite different and as such the category itself is quite different.…”

“…cit. When the ground field is algebraically closed, the construction was completely generalised to all coalgebras by Chuang, Lazarev, and Mannan [5]. The authors therein chose to work in the dual setting of pseudo-compact unital commutative differential graded algebras.…”

“…The Koszul duality is also extended in [5]. The Koszul dual therein is the category of formal coproducts of curved Lie algebras.…”

“…Sections 2 and 3 recall (without proof) the necessary facts concerning the category of formal products of a given category, the extended Hinich category, and the category of pseudo-compact unital commutative differential graded algebras. For more details see the original paper [5].…”