Idempotent Symmetries in Algebra and Topology
The proofs of Ravenel’s conjectures  and their reinterpretations in the form of the classification of the homotopy idempotent functors of spectra commuting with telescopes by Devintaz–Hopkins–Smith [7, 12], were a culmination of a few decades of progress achieved in stable homotopy theory. The simplicity of this classification is remarkable. For each prime p, the category of restrictions of these functors to p-local finite spectra is isomorphic to the poset of natural numbers. The obtained invariant is called the Morava–Hopkins type. The stable classification was generalized to the classification of so-called Bousfield localizations of finite p-local spaces; see Bousfield . This unstable classification is also remarkably simple. Bousfield showed that, in addition to the Morava–Hopkins type invariant, connectivity determines such functors.
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