Higher Koszul duality and $n$-affineness

We study ${\mathbb{E}}_n$-Koszul duality for pairs of algebras of the form ${\mathrm{C}}_{\bullet }({\mathrm{\Omega }}_{\ast }^{n}X;\mathbb{k})\leftrightarrow {\mathrm{C}}^{\bullet }(X;\mathbb{k})$, and the closely related question of $n$-affineness for Betti stacks. It was expected, but not known, that ${\mathbb{E}}_n$-Koszul duality should induce a kind of Morita equivalence between categories of iterated modules. We establish this rigorously by proving that the $(\mathrm{\infty },n)$-category of iterated modules over ${\mathrm{C}}_{\bullet }({\mathrm{\Omega }}_{\ast }^{n+1}X;\mathbb{k})$ is equivalent to the $(\mathrm{\infty },n)$-category of quasi-coherent sheaves of $(\mathrm{\infty },n-1)$-categories on $\mathrm{c}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}({\mathrm{C}}^{\bullet }(X;\mathbb{k}))$, where $\mathrm{c}\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}({\mathrm{C}}^{\bullet }(X;\mathbb{k}))$ is the cospectrum of ${\mathrm{C}}^{\bullet }(X;\mathbb{k})$. By the monodromy equivalence, these categories are also equivalent to the category of higher local systems on $X$, $n{\mathbf{L}\mathbf{o}\mathbf{c}\mathbf{S}\mathbf{y}\mathbf{s}\mathbf{C}\mathbf{a}\mathbf{t}}^{n-1}(X;\mathbb{k})$. 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.