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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.09150 |
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| _version_ | 1866909610300932096 |
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| author | Li, Yifan |
| author_facet | Li, Yifan |
| contents | In this article, we give an introduction to the notion of ambidexterity and norm map, and construct inductively the canonical norm map for $m$-truncated maps for some $m\geq-1$, on which the definitions of integration and cardinality are built. We then use several propositions to justify the properties of cardinality and integration and their compatibility with monoidal structure. We give a brief introduction of the definition and behaviors of semiadditive height. Focusing on stable monoidal $p$-local $\infty$-categories of height 1, for any finite group $G$, with the help of Möbius function and Burnside ring, we give an explicit decomposition of the cardinality of $BG$ into an expression of the cardinality of $BC_p$. Eventually, we generalize the result and conclude with a formula of the cardinality of any $π$-finite space $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09150 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cardinalities in Height 1 Li, Yifan Algebraic Topology 18N60, 55P42 In this article, we give an introduction to the notion of ambidexterity and norm map, and construct inductively the canonical norm map for $m$-truncated maps for some $m\geq-1$, on which the definitions of integration and cardinality are built. We then use several propositions to justify the properties of cardinality and integration and their compatibility with monoidal structure. We give a brief introduction of the definition and behaviors of semiadditive height. Focusing on stable monoidal $p$-local $\infty$-categories of height 1, for any finite group $G$, with the help of Möbius function and Burnside ring, we give an explicit decomposition of the cardinality of $BG$ into an expression of the cardinality of $BC_p$. Eventually, we generalize the result and conclude with a formula of the cardinality of any $π$-finite space $A$. |
| title | Cardinalities in Height 1 |
| topic | Algebraic Topology 18N60, 55P42 |
| url | https://arxiv.org/abs/2505.09150 |