I will discuss an infinity-category obtained from that of pointed spaces by inverting the maps inducing isomorphisms in vn-periodic homotopy groups. The case n=0 corresponds to rational homotopy theory. In analogy with Quillen’s results in the rational case, I will outline how this vn-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in T(n)-local spectra (or a variant for K(n)-local spectra). One can also compare it to the homotopy theory of cocommutative coalgebras in T(n)-local spectra, where there is only an equivalence up to a certain "Goodwillie convergence" issue. I will describe the relevant operadic and cooperadic structures and a form of Koszul duality relevant to this setting.