Lawvere completeness in Topology

TL;DR

This paper extends Lawvere's notion of completeness to a broader class of categories, revealing its significance for topological and quasi-uniform spaces, and introduces a canonical structure that enables a Yoneda embedding.

Contribution

It introduces Lawvere completeness for $(bT,bV)$-categories and explores its implications for topology and quasi-uniform spaces, including a canonical structure and Yoneda embedding.

Findings

Lawvere completeness corresponds to weak sobriety in topological spaces.

It aligns with Cauchy completeness in quasi-uniform spaces.

A canonical $(bT,bV)$-category structure on $bV$ is established.

Abstract

It is known since 1973 that Lawvere's notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for $(\mathbb{T},\mathsf{V})$-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that $\mathsf{V}$ has a canonical $(\mathbb{T},\mathsf{V})$-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of $(\mathbb{T},\mathsf{V})$-categories.