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I will discuss an infinity-category obtained from that of pointed spaces by inverting the maps inducing isomorphisms in $v_n$-periodic homotopy groups. The case $n = 0$ corresponds to rational homotopy theory. In analogy with Quillenâ s results in the rational case, I will outline how this $v_n$-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.