We consider a homotopy theory 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 Quillen's results in the rational case, we prove that this $v_n$-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in $T(n)$-local spectra. We also compare it to the homotopy theory of commutative coalgebras in $T(n)$-local spectra, where it turns out there is only an equivalence up to a certain convergence issue of the Goodwillie tower of the identity.