Higher Koszul duality and n-affineness

We study Eₙ-Koszul duality for pairs of algebras of the form C_ (Ω^n_*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, nLocSysCat^n-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 π₍+₁ (X) is an obstruction to n-affineness.