Monadicity of the Bousfield-Kuhn functor

TL;DR

This paper proves that localizing the infinity-category of spaces at v_n-periodic equivalences for n≥1 is equivalent to algebras over a specific monad derived from the Bousfield--Kuhn functor, extending rational homotopy theory.

Contribution

It establishes the monadicity of the Bousfield--Kuhn functor in the context of v_n-periodic localization for n≥1.

Findings

Localization at v_n-equivalences is monadic for n≥1.

The monad is constructed from the Bousfield--Kuhn functor.

Extends rational homotopy theory to higher chromatic levels.

Abstract

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