Metric enrichment, finite generation, and the path comonad

TL;DR

This paper explores categories enriched over complete metric spaces, establishing their structural properties, and characterizes isometry-generated objects, contributing to the understanding of metric-enriched category theory.

Contribution

It proves that the category of intrinsic complete metric spaces is locally presentable, closed monoidal, and comonadic over CMet, and characterizes isometry-generated objects in several metric categories.

Findings

The category CPMet is locally $eth_1$-presentable, closed monoidal, and comonadic over CMet.

The category CCMet is not closed monoidal.

Characterization of isometry-$eth_0$-generated objects in CMet, CPMet, and CCMet.

Abstract

We prove a number of results involving categories enriched over \textsc{CMet}, the category of complete metric spaces with possibly infinite distances. The category \textsc{CPMet} of intrinsic complete metric spaces is locally $\aleph_1$-presentable, closed monoidal, and comonadic over \textsc{CMet}. We also prove that the category \textsc{CCMet} of convex complete metric spaces is not closed monoidal and characterize the isometry-$\aleph_0$-generated objects in \textsc{CMet}, \textsc{CPMet} and \textsc{CCMet}, answering questions by Di Liberti and Rosick\'{y}. Other results include the automatic completeness of a colimit of bi-Lipschitz morphisms of complete metric spaces and a characterization of those pairs (metric space, unital $C^*$-algebra) that have a tensor product in the \textsc{CMet}-enriched category of unital $C^*$-algebras.