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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.07869 |
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| _version_ | 1866911049414868992 |
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| author | Mateo, Adrián Doña |
| author_facet | Mateo, Adrián Doña |
| contents | In the paper where he defined the Cauchy completion of a $\mathscr{V}$-category, Lawvere also defined a condition on a $\mathscr{V}$-functor which made it analogous to a map of metric spaces whose image is topologically dense in its codomain. We call this condition Cauchy density. In this note, we focus on the fully faithful Cauchy dense $\mathscr{V}$-functors, and show that the Cauchy completion of $\mathscr{A}$ is the largest $\mathscr{V}$-category that admits a fully faithful Cauchy dense $\mathscr{V}$-functor from $\mathscr{A}$. Moreover, we show that $F \colon \mathscr{A} \to \mathscr{B}$ is fully faithful and Cauchy dense iff $[F,\mathscr{C}] \colon [\mathscr{B},\mathscr{C}] \to [\mathscr{A},\mathscr{C}]$ is an equivalence for any Cauchy complete $\mathscr{C}$. Finally, we provide examples and characterisations of Cauchy dense functors in various contexts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07869 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cauchy density Mateo, Adrián Doña Category Theory 18A22 (Primary), 18D20, 18D60 (Secondary) In the paper where he defined the Cauchy completion of a $\mathscr{V}$-category, Lawvere also defined a condition on a $\mathscr{V}$-functor which made it analogous to a map of metric spaces whose image is topologically dense in its codomain. We call this condition Cauchy density. In this note, we focus on the fully faithful Cauchy dense $\mathscr{V}$-functors, and show that the Cauchy completion of $\mathscr{A}$ is the largest $\mathscr{V}$-category that admits a fully faithful Cauchy dense $\mathscr{V}$-functor from $\mathscr{A}$. Moreover, we show that $F \colon \mathscr{A} \to \mathscr{B}$ is fully faithful and Cauchy dense iff $[F,\mathscr{C}] \colon [\mathscr{B},\mathscr{C}] \to [\mathscr{A},\mathscr{C}]$ is an equivalence for any Cauchy complete $\mathscr{C}$. Finally, we provide examples and characterisations of Cauchy dense functors in various contexts. |
| title | Cauchy density |
| topic | Category Theory 18A22 (Primary), 18D20, 18D60 (Secondary) |
| url | https://arxiv.org/abs/2507.07869 |