Koszul duality and a conjecture of Francis-Gaitsgory

TL;DR

This paper investigates the scope of Koszul duality, disproves a conjecture by Francis-Gaitsgory, and establishes an equivalence between specific subcategories of algebras and coalgebras, expanding understanding of their duality.

Contribution

It demonstrates that Koszul duality equates nilcomplete algebras with conilcomplete coalgebras, refuting Francis-Gaitsgory's conjecture and providing a new framework for known results.

Findings

Disproves Francis-Gaitsgory's conjecture.

Establishes equivalence between nilcomplete algebras and conilcomplete coalgebras.

Unifies previous partial results on Koszul duality.

Abstract

Koszul duality is a fundamental correspondence between algebras for an operad $\mathcal{O}$ and coalgebras for its dual cooperad $B\mathcal{O}$, built from $\mathcal{O}$ using the bar construction. Francis-Gaitsgory proposed a conjecture about the general behavior of this duality. The main result of this paper, roughly speaking, is that Koszul duality provides an equivalence between the subcategories of nilcomplete algebras and conilcomplete coalgebras and that these are the largest possible subcategories for which such a result holds. This disproves Francis-Gaitsgory's prediction, but does provide an adequate replacement. We show that many previously known partial results about Koszul duality can be deduced from our results.