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We consider the localization of the $\infty$-category of spaces at the$v_n$-periodic equivalences, the case $n=0$ being rational homotopy theory. Weprove that this localization is for $n\geq 1$ equivalent to algebras over acertain monad on the $\infty$-category of $T(n)$-local spectra. This monad isbuilt from the Bousfield--Kuhn functor.

Monadicity of the Bousfield-Kuhn functor