Higher Koszul duality and n-affineness

We study Eₙ-Koszul duality for pairs of algebras of the form C•(Ωⁿ_*X;) ↔ C•(X;), and the closely related question of n-affineness for Betti stacks. It was expected, but not known, that Eₙ-Koszul duality should induce a kind of Morita equivalence between categories of iterated modules. We establish this rigorously by proving that the (∞,n)-category of iterated modules over C•(Ω_*ⁿ⁺¹X;) is equivalent to the (∞,n)-category of quasi-coherent sheaves of (∞,n-1)-categories on cSpec(C•(X;)), where cSpec(C•(X;)) is the cospectrum of C•(X;). By the monodromy equivalence, these categories are also equivalent to the category of higher local systems on X, nLocSysCatⁿ⁻¹(X;). Our result is new already in the classical case $n=1$, although it can be seen to recover well known formulations of E₁-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 πₙ₊₁(X) is an obstruction to n-affineness.