Globalizing and stabilizing global $\infty$-categories
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909061553848320 |
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| author | Linskens, Sil |
| author_facet | Linskens, Sil |
| contents | We consider the question of cocompleting partially presentable parametrized $\infty$-categories in the sense of arXiv:2307.11001. As our main result we show that in certain cases one may compute such relative cocompletions via a very explicit formula given in terms of partially lax limits. We then apply this to equivariant homotopy theory, building on the work of op. cit. and arXiv:2301.08240, to conclude that the global $\infty$-category of globally equivariant spectra is the relative cocompletion of the global $\infty$-category of equivariant spectra. Evaluating at a group $G$ we obtain a description of the $\infty$-category of $G$-global spectra as a partially lax limit, extending the main result of arXiv:2206.01556 for finite groups to $G$-global homotopy theory. Finally we investigate the question of stabilizing global $\infty$-categories by inverting the action of representation spheres, and deduce a second universal property for the global $\infty$-category of globally equivariant spectra, similar to that of arXiv:2302.06207. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02264 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Globalizing and stabilizing global $\infty$-categories Linskens, Sil Algebraic Topology Category Theory We consider the question of cocompleting partially presentable parametrized $\infty$-categories in the sense of arXiv:2307.11001. As our main result we show that in certain cases one may compute such relative cocompletions via a very explicit formula given in terms of partially lax limits. We then apply this to equivariant homotopy theory, building on the work of op. cit. and arXiv:2301.08240, to conclude that the global $\infty$-category of globally equivariant spectra is the relative cocompletion of the global $\infty$-category of equivariant spectra. Evaluating at a group $G$ we obtain a description of the $\infty$-category of $G$-global spectra as a partially lax limit, extending the main result of arXiv:2206.01556 for finite groups to $G$-global homotopy theory. Finally we investigate the question of stabilizing global $\infty$-categories by inverting the action of representation spheres, and deduce a second universal property for the global $\infty$-category of globally equivariant spectra, similar to that of arXiv:2302.06207. |
| title | Globalizing and stabilizing global $\infty$-categories |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2401.02264 |