Cauchy density

TL;DR

This paper explores the properties of Cauchy dense functors in enriched category theory, characterizing Cauchy completion as a maximal construction and providing criteria for functor equivalences in Cauchy complete contexts.

Contribution

It characterizes fully faithful Cauchy dense functors and shows that Cauchy completion is the largest such extension, with new criteria for functor equivalences in enriched categories.

Findings

Cauchy completion is the largest category admitting a fully faithful Cauchy dense functor.

Fully faithful Cauchy dense functors induce equivalences of functor categories.

Examples and characterizations of Cauchy dense functors in various contexts are provided.

Abstract

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.