Parametrized (higher) semiadditivity and the universality of spans
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918032152985600 |
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| author | Cnossen, Bastiaan Lenz, Tobias Linskens, Sil |
| author_facet | Cnossen, Bastiaan Lenz, Tobias Linskens, Sil |
| contents | Semiadditivity of an $\infty$-category, i.e. the existence of biproducts, provides it with useful algebraic structure in the form of a canonical enrichment in commutative monoids. This ultimately comes from the fact that the $\infty$-category of commutative monoids is the universal semiadditive $\infty$-category equipped with a finite-product-preserving functor to spaces, or equivalently that the $(2,1)$-category of spans of finite sets is the universal semiadditive $\infty$-category. In this article, we prove a vast generalization of these facts in the context of parametrized semiadditivity, a notion we define using Hopkins-Lurie's framework of ambidexterity. This simultaneously generalizes a result of Harpaz for higher semiadditivity and a result of Nardin for equivariant semiadditivity. We deduce that every parametrized semiadditive $\infty$-category is canonically enriched in Mackey functors/sheaves with transfers.
As an application, we reprove the Mackey functor description of global spectra first obtained by the second-named author and generalize it to $G$-global spectra. Moreover, we obtain universal characterizations of the $\infty$-categories of $\mathbb Z$-valued $G$-Mackey profunctors and of quasi-finitely genuine $G$-spectra as studied by Kaledin and Krause-McCandless-Nikolaus, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07676 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Parametrized (higher) semiadditivity and the universality of spans Cnossen, Bastiaan Lenz, Tobias Linskens, Sil Algebraic Topology 55P91, 18N60 (Primary) Semiadditivity of an $\infty$-category, i.e. the existence of biproducts, provides it with useful algebraic structure in the form of a canonical enrichment in commutative monoids. This ultimately comes from the fact that the $\infty$-category of commutative monoids is the universal semiadditive $\infty$-category equipped with a finite-product-preserving functor to spaces, or equivalently that the $(2,1)$-category of spans of finite sets is the universal semiadditive $\infty$-category. In this article, we prove a vast generalization of these facts in the context of parametrized semiadditivity, a notion we define using Hopkins-Lurie's framework of ambidexterity. This simultaneously generalizes a result of Harpaz for higher semiadditivity and a result of Nardin for equivariant semiadditivity. We deduce that every parametrized semiadditive $\infty$-category is canonically enriched in Mackey functors/sheaves with transfers. As an application, we reprove the Mackey functor description of global spectra first obtained by the second-named author and generalize it to $G$-global spectra. Moreover, we obtain universal characterizations of the $\infty$-categories of $\mathbb Z$-valued $G$-Mackey profunctors and of quasi-finitely genuine $G$-spectra as studied by Kaledin and Krause-McCandless-Nikolaus, respectively. |
| title | Parametrized (higher) semiadditivity and the universality of spans |
| topic | Algebraic Topology 55P91, 18N60 (Primary) |
| url | https://arxiv.org/abs/2403.07676 |