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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.15887 |
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| _version_ | 1866916297763192832 |
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| author | Doni, Matteo |
| author_facet | Doni, Matteo |
| contents | We establish the feasibility of investigating the theory of $R\text{-}\mathrm{Mod}$-enriched categories, for any commutative and unitary ring $R$, through the framework of $\mathbb{A}\mathrm{b}$-enriched category theory. In particular, we prove that the category of $R$-$\mathrm{Mod}$-enriched categories, $Cat(R$-$\mathrm{Mod})$, the category of $\underline{R}$-modules inside $Cat(\mathbb{A}\mathrm{b})$, $\mathrm{LMod}_{\underline{R}}(Cat(\mathbb{A}\mathrm{b}))$, and the category of $Cat(\mathbb{A}\mathrm{b})$-enriched functors, $Fun^{Cat(\mathbb{A}\mathrm{b})}(\underline{\underline{R}},Cat(\mathbb{A}\mathrm{b}))$ are equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15887 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $R\text{-}\mathrm{Mod}$-enriched categories are left $\underline{R}$-module objects of $Cat(\mathbb{A}\mathrm{b})$ and $Cat(\mathbb{A}\mathrm{b})$-enriched functors Doni, Matteo Category Theory We establish the feasibility of investigating the theory of $R\text{-}\mathrm{Mod}$-enriched categories, for any commutative and unitary ring $R$, through the framework of $\mathbb{A}\mathrm{b}$-enriched category theory. In particular, we prove that the category of $R$-$\mathrm{Mod}$-enriched categories, $Cat(R$-$\mathrm{Mod})$, the category of $\underline{R}$-modules inside $Cat(\mathbb{A}\mathrm{b})$, $\mathrm{LMod}_{\underline{R}}(Cat(\mathbb{A}\mathrm{b}))$, and the category of $Cat(\mathbb{A}\mathrm{b})$-enriched functors, $Fun^{Cat(\mathbb{A}\mathrm{b})}(\underline{\underline{R}},Cat(\mathbb{A}\mathrm{b}))$ are equivalent. |
| title | $R\text{-}\mathrm{Mod}$-enriched categories are left $\underline{R}$-module objects of $Cat(\mathbb{A}\mathrm{b})$ and $Cat(\mathbb{A}\mathrm{b})$-enriched functors |
| topic | Category Theory |
| url | https://arxiv.org/abs/2406.15887 |