  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.
