Cauchy-completions and the rule of unique choice in relational doctrines

Abstract

Lawvere's generalised the notion of complete metric space to the field of enriched categories: an enriched category is said to be Cauchy-complete if every left adjoint bimodule into it is represented by an enriched functor. Looking at this definition from a logical standpoint, regarding bimodules as an abstraction of relations and functors as an abstraction of functions, Cauchy-completeness resembles a formulation of the rule of unique choice. In this paper, we make this analogy precise, using the language of relational doctrines, a categorical tool that provides a functorial description of the calculus of relations, in the same way Lawvere's hyperdoctrines give a functorial description of predicate logic. Given a relational doctrine, we define Cauchy-complete objects as those objects of the domain category satisfying the rule of unique choice. Then, we present a universal construction that completes a relational doctrine with the rule of unique choice, that is, producing a new relational doctrine where all objects are Cauchy-complete. We also introduce relational doctrines with singleton objects and show that these have the minimal structure needed to build the reflector of the full subcategory of its domain on Cauchy-complete objects. The main result is that this reflector exists if and only if the relational doctrine has singleton objects and this happens if and only if its restriction to Cauchy-complete objects is equivalent to its completion with the rule of unique choice. We support our results with many examples, also falling outside the scope of standard doctrines, such as complete metric spaces, Banach spaces and compact Hausdorff spaces in the general context of monoidal topology, which are all shown to be Cauchy-complete objects for appropriate relational doctrines.