Universal property of the Bousfield--Kuhn functor
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917793100726272 |
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| author | Shi, Yuqing |
| author_facet | Shi, Yuqing |
| contents | We present a universal property of the Bousfield--Kuhn functor $\operatornameΦ_h$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_01116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Universal property of the Bousfield--Kuhn functor Shi, Yuqing Algebraic Topology 55Q51, 16S30, 18N60 (Primary) 18N70, 55U35, 18M70 (Secondary) We present a universal property of the Bousfield--Kuhn functor $\operatornameΦ_h$ 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. |
| title | Universal property of the Bousfield--Kuhn functor |
| topic | Algebraic Topology 55Q51, 16S30, 18N60 (Primary) 18N70, 55U35, 18M70 (Secondary) |
| url | https://arxiv.org/abs/2410.01116 |