Higher Koszul duality and $n$-affineness

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 $\mathrm{C}^{\bullet}(X;\Bbbk)$ 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 $\pi_{n+1}(X)$ is an obstruction to $n$-affineness.