Categorical spectra as pointed $(\infty,\mathbb{Z})$-categories
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866917827984752640 |
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| author | Kern, David |
| author_facet | Kern, David |
| contents | Lessard's $\mathbb{Z}$-categories are an analogue of $ω$-categories possessing cells in all positive and negative dimensions. Categorical spectra, developed by Stefanich, are an analogue of spectra obtained by replacing the suspension of pointed $\infty$-groupoids by that of pointed $(\infty,ω)$-categories. We give an $\infty$-categorical definition of weak $\mathbb{Z}$-categories (alias $(\infty,\mathbb{Z})$-categories), and show categorical spectra to be equivalent to pointed $(\infty,\mathbb{Z})$-categories. In particular, we show that the stable cells of categorical spectra coincide with the natural cells of $(\infty,\mathbb{Z})$-categories, and recover Lessard's description of spectra as pointed weak $\mathbb{Z}$-groupoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_02578 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Categorical spectra as pointed $(\infty,\mathbb{Z})$-categories Kern, David Category Theory Algebraic Topology 18N65, 55P42 Lessard's $\mathbb{Z}$-categories are an analogue of $ω$-categories possessing cells in all positive and negative dimensions. Categorical spectra, developed by Stefanich, are an analogue of spectra obtained by replacing the suspension of pointed $\infty$-groupoids by that of pointed $(\infty,ω)$-categories. We give an $\infty$-categorical definition of weak $\mathbb{Z}$-categories (alias $(\infty,\mathbb{Z})$-categories), and show categorical spectra to be equivalent to pointed $(\infty,\mathbb{Z})$-categories. In particular, we show that the stable cells of categorical spectra coincide with the natural cells of $(\infty,\mathbb{Z})$-categories, and recover Lessard's description of spectra as pointed weak $\mathbb{Z}$-groupoids. |
| title | Categorical spectra as pointed $(\infty,\mathbb{Z})$-categories |
| topic | Category Theory Algebraic Topology 18N65, 55P42 |
| url | https://arxiv.org/abs/2410.02578 |