Exponentiable Grothendieck categories in flat Algebraic Geometry
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866908475709194240 |
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| author | Di Liberti, Ivan González, Julia Ramos |
| author_facet | Di Liberti, Ivan González, Julia Ramos |
| contents | We introduce and describe the $2$-category $\mathsf{Grt}_{\flat}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $\boxtimes$ restricts nicely to $\mathsf{Grt}_{\flat}$. Then, we characterize exponentiable objects with respect to $\boxtimes$: these are continuous Grothendieck categories. In particular, locally finitely presentable Grothendieck categories are exponentiable. Consequently, we have that, for a quasi-compact quasi-separated scheme $X$, the category of quasi-coherent sheaves $\mathsf{Qcoh}(X)$ is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2103_07876 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Exponentiable Grothendieck categories in flat Algebraic Geometry Di Liberti, Ivan González, Julia Ramos Algebraic Geometry Category Theory 14A22, 14F06, 18B25, 18C35, 18F10, 18M05, 18E10 We introduce and describe the $2$-category $\mathsf{Grt}_{\flat}$ of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories $\boxtimes$ restricts nicely to $\mathsf{Grt}_{\flat}$. Then, we characterize exponentiable objects with respect to $\boxtimes$: these are continuous Grothendieck categories. In particular, locally finitely presentable Grothendieck categories are exponentiable. Consequently, we have that, for a quasi-compact quasi-separated scheme $X$, the category of quasi-coherent sheaves $\mathsf{Qcoh}(X)$ is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials. |
| title | Exponentiable Grothendieck categories in flat Algebraic Geometry |
| topic | Algebraic Geometry Category Theory 14A22, 14F06, 18B25, 18C35, 18F10, 18M05, 18E10 |
| url | https://arxiv.org/abs/2103.07876 |