Higher Koszul duality and $n$-affineness

Abstract

In this paper we study ${\mathbb{E}}_n$-Koszul duality in the topological setting, and the closely related question of \emph{$n$-affineness} for Betti stacks. The ${\mathbb{E}}_n$-Koszul dual of the algebra of chains on the $n$-fold loop space of a space $X$ is the algebra of cochains on $X$. It was expected that ${\mathbb{E}}_n$-Koszul duality should induce a kind of Morita equivalence between categories of iterated modules, but even the precise formulation of such a statement was not known. We give a rigorous formulation, and a proof, of such an ${\mathbb{E}}_n$-Koszul duality in the topological setting as an equivalence of $(\infty ,n)$-categories. Conceptually, our main innovation is highlighting the coaffine stack defined by the \emph{cospectrum} of ${\mathrm{C}}^{\bullet }(X;\mathbb{k})$ as a key geometric object supporting Koszul duality. Our result is new already in the classical case $n=1$, although it can be seen to recover well known formulations of ${\mathbb{E}}_1$-Koszul duality as a Morita equivalence of module categories (up to appropriate completions of the $t$-structures). We also investigate (higher) affineness properties of Betti stacks. We give a complete characterization of $n$-affine Betti stacks, in terms of the $0$-affineness of their iterated loop space. As a consequence, we prove that $n$-truncated Betti stacks are $n$-affine; and that $\ddot{I}{€}_{n+1}(X)$ is an obstruction to $n$-affineness.

Fixed a gap in the proof of the main theorem; streamlined the exposition and added some further results. arXiv admin note: substantial text overlap with arXiv:2501.10241