Monadicity of the Bousfield-Kuhn functor

We consider the localization of the $\mathrm{\infty }$-category of spaces at the $v_n$-periodic equivalences, the case $n=0$ being rational homotopy theory. We prove that this localization is for $n\ge 1$ equivalent to algebras over a certain monad on the $\mathrm{\infty }$-category of $T(n)$-local spectra. This monad is built from the Bousfield–Kuhn functor.