Cauchy density

Abstract

In the paper where he defined the Cauchy completion of a $\mathcal{V}$-category, Lawvere also defined a condition on a $\mathcal{V}$-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 $\mathcal{V}$-functors, and show that the Cauchy completion of $\mathcal{A}$ is the largest $\mathcal{V}$-category that admits a fully faithful Cauchy dense $\mathcal{V}$-functor from $\mathcal{A}$. Moreover, we show that $F:\mathcal{A}\to \mathcal{B}$ is fully faithful and Cauchy dense iff $[F,\mathcal{C}]:[\mathcal{B},\mathcal{C}]\to [\mathcal{A},\mathcal{C}]$ is an equivalence for any Cauchy complete $\mathcal{C}$. Finally, we provide examples and characterisations of Cauchy dense functors in various contexts.

15 pages