Universal property of the Bousfield--Kuhn functor

TL;DR

This paper establishes a universal property of the Bousfield--Kuhn functor at any height, linking $v_h$-periodic homotopy types with $ ext{T}(h)$-local spectra through costabilisation and spectral Lie algebra models.

Contribution

It proves a universal property for the Bousfield--Kuhn functor for all heights by relating costabilisations of homotopy types and spectra using spectral Lie algebra models.

Findings

Costabilisation of $v_h$-periodic homotopy types is equivalent to $ ext{T}(h)$-local spectra.

Spectral Lie algebra models effectively describe $v_h$-periodic homotopy types.

Higher enveloping algebras connect spectral Lie algebras with $ ext{E}_n$-algebras.

Abstract

We present a universal property of the Bousfield--Kuhn functor $\operatorname{\Phi}_h$ of height $h$, for every positive natural number $h$. This result is achieved by proving that the costabilisation of the $\infty$-category of $v_h$-periodic homotopy types is equivalent to the $\infty$-category of $\operatorname{T}(h)$-local spectra. A key component in our proofs is the spectral Lie algebra model for $v_h$-periodic homotopy types (see arXiv:1803.06325): We relate the costabilisation of the $\infty$-category of spectral Lie algebras with the costabilisations of the $\infty$-category of non-unital $\mathcal{E}_{n}$-algebras, via our construction of higher enveloping algebras of spectral Lie algebras.