Monadicity of the Bousfield–Kuhn functor

Abstract

Let M f n be the localization of the ∞-category of spaces at the vnperiodic equivalences, the case n = 0 being rational homotopy theory. We prove that M f n is for n ≥ 1 equivalent to algebras over a certain monad on the ∞-category of T (n)-local spectra. This monad is built from the Bousfield-Kuhn functor.

Lie algebras and $v_n$-periodic spaces

“…In joint work with Eldred, Mathew, and Meier [20] we prove the following theorem. We include a sketch of the proof for the reader's convenience.…”

“…The two adjunctions above offer complementary perspectives on the ∞category S vn . In joint work with Eldred, Mathew, and Meier [20] we prove that the adjoint pair (Θ, Φ) is monadic, meaning that Φ gives an equivalence between S vn and the ∞-category of algebras for the monad ΦΘ on Sp T (n) . Here we go further and explicitly identify this monad as the free Lie algebra monad.…”

“…By Corollary 5.5.8.17 of [36] it suffices to show that Φ preserves filtered colimits and geometric realizations. The first is Proposition 3.21, and xthe second was part of the proof of Theorem 4.1 above (which first appeared as Proposition 4.1 in [20]).…”

“…preserves geometric realizations, since the connectivity d n+1 of the spaces involved exceeds the dimension of V (see Proposition 4.2 of [20]). This finishes the proof.…”

“…I have benefitted much from reading their work, as well as from several inspiring talks by and conversations with Mark Behrens. Theorem 2.6 (the comparison with Lie algebras) offers a different (and sharper) perspective on v n -periodic homotopy theory, which builds on joint work with Rosona Eldred, Akhil Mathew, and Lennart Meier [20] carried out at the Hausdorff Research Institute for Mathematics. I wish to thank my collaborators for an inspiring semester and the Institute for its hospitality and excellent working conditions.…”