2025/07/10 by Adrián Doña Mateo, Mateo, Adrián Doña
Computer Science · Mathematics · #18A22 (Primary) #18D20 #18D60 (Secondary) #Advanced Topology and Set Theory #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.2507.07869
openalex publication_date 2025/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the paper where he defined the Cauchy completion of a \mathscrV-category, Lawvere also defined a condition on a \mathscrV-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 \mathscrV-functors, and show that the Cauchy completion of \mathscrA is the largest \mathscrV-category that admits a fully faithful Cauchy dense \mathscrV-functor from \mathscrA. Moreover, we show that F \colon \mathscrA → \mathscrB is fully faithful and Cauchy dense iff [F,\mathscrC] \colon [\mathscrB,\mathscrC] → [\mathscrA,\mathscrC] is an equivalence for any Cauchy complete \mathscrC. Finally, we provide examples and characterisations of Cauchy dense functors in various contexts.