Global homotopy theory via partially lax limits
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866911006682251264 |
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| author | Linskens, Sil Nardin, Denis Pol, Luca |
| author_facet | Linskens, Sil Nardin, Denis Pol, Luca |
| contents | We provide new $\infty$-categorical models for unstable and stable global homotopy theory. We use the notion of partially lax limits to formalize the idea that a global object is a collection of $G$-objects, one for each compact Lie group $G$, which are compatible with the restriction-inflation functors. More precisely, we show that the $\infty$-category of global spaces is equivalent to a partially lax limit of the functor sending a compact Lie group $G$ to the $\infty$-category of $G$-spaces. We also prove the stable version of this result, showing that the $\infty$-category of global spectra is equivalent to the partially lax limit of a diagram of $G$-spectra. Finally, the techniques employed in the previous cases allow us to describe the $\infty$-category of proper $G$-spectra for a Lie group $G$, as a limit of a diagram of $H$-spectra for $H$ running over all compact subgroups of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_01556 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Global homotopy theory via partially lax limits Linskens, Sil Nardin, Denis Pol, Luca Algebraic Topology Category Theory We provide new $\infty$-categorical models for unstable and stable global homotopy theory. We use the notion of partially lax limits to formalize the idea that a global object is a collection of $G$-objects, one for each compact Lie group $G$, which are compatible with the restriction-inflation functors. More precisely, we show that the $\infty$-category of global spaces is equivalent to a partially lax limit of the functor sending a compact Lie group $G$ to the $\infty$-category of $G$-spaces. We also prove the stable version of this result, showing that the $\infty$-category of global spectra is equivalent to the partially lax limit of a diagram of $G$-spectra. Finally, the techniques employed in the previous cases allow us to describe the $\infty$-category of proper $G$-spectra for a Lie group $G$, as a limit of a diagram of $H$-spectra for $H$ running over all compact subgroups of $G$. |
| title | Global homotopy theory via partially lax limits |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2206.01556 |