For a simply connected and finite CW complex, its algebra of cochains with coefficients over a field k of characteristic 0 is Koszul dual (as an E_1-algebra) to the algebra of k-chains on its based loop space. This correspondence produces an equivalence of categories between modules over the former and ind-coherent modules over the latter. When a CW complex is n-connected, one can phrase an analogous E_n-Koszul duality statement relating its algebra of k-cochains and the algebra of k-chains over its n-fold based loop space. However, this duality cannot be interpreted as an equivalence between different categories of modules anymore. In this talk, I propose a way to generalize the correspondence between modules over Koszul dual E_1-algebras to E_n-algebras, via the formalism of iterated modules and presentable higher (n-)categories. This is a joint work in progress with J. Pascaleff and N. Sibilla.