Koszul duality and a conjecture of Francis-Gaitsgory

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.