A symmetric monoidal fracture square
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913574393217024 |
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| author | Naumann, Niko Pol, Luca Ramzi, Maxime |
| author_facet | Naumann, Niko Pol, Luca Ramzi, Maxime |
| contents | Given a symmetric monoidal stable $\infty$-category $\mathcal{C}$ which is rigidly-compactly generated and a set of compact objects $\mathcal{K}$ of $\mathcal{C}$, one can form the subcategories of $\mathcal{K}$-complete and $\mathcal{K}$-local objects. The goal of this paper is to explain how to recover $\mathcal{C}$ from its $\mathcal{K}$-local and $\mathcal{K}$-complete subcategories while retaining the symmetric monoidal structure. Specializing to the case where $\mathcal{C}$ is the $\infty$-category of $G$-spectra for a finite group $G$, our result can be viewed as a symmetric monoidal variant of the isotropy separation decomposition, a version of which appeared previously in work of Krause. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A symmetric monoidal fracture square Naumann, Niko Pol, Luca Ramzi, Maxime Algebraic Topology Algebraic Geometry Category Theory Given a symmetric monoidal stable $\infty$-category $\mathcal{C}$ which is rigidly-compactly generated and a set of compact objects $\mathcal{K}$ of $\mathcal{C}$, one can form the subcategories of $\mathcal{K}$-complete and $\mathcal{K}$-local objects. The goal of this paper is to explain how to recover $\mathcal{C}$ from its $\mathcal{K}$-local and $\mathcal{K}$-complete subcategories while retaining the symmetric monoidal structure. Specializing to the case where $\mathcal{C}$ is the $\infty$-category of $G$-spectra for a finite group $G$, our result can be viewed as a symmetric monoidal variant of the isotropy separation decomposition, a version of which appeared previously in work of Krause. |
| title | A symmetric monoidal fracture square |
| topic | Algebraic Topology Algebraic Geometry Category Theory |
| url | https://arxiv.org/abs/2411.05467 |