$\mathscr Q$-Sets and Friends: Regarding Singleton and Gluing Completeness

TL;DR

This paper extends the concept of $ ext{ extsc{Higgs}}$'s $ ext{ extsc{ extOmega}}$-sets to quantales, studying various notions of completeness, their relations, and categorical properties within the framework of $ ext{ extsc{ extbf{Q}}}$-sets.

Contribution

It introduces new categorical characterizations of singleton completeness and establishes their equivalence with functional morphisms, expanding the theory of $ ext{ extsc{ extbf{Q}}}$-sets.

Findings

Singleton complete $ ext{ extsc{ extbf{Q}}}$-sets are equivalent to functional morphisms.

The categorical inclusion of singleton complete $ ext{ extsc{ extbf{Q}}}$-sets creates limits.

Multiple notions of completeness are related through completion functors and reflective subcategories.

Abstract

This work is largely focused on extending D. Higgs' $\Omega$-sets to the context of quantales, following the broad program of U. H\"ohle, we explore the rich category of $\mathscr Q$-sets for strong, integral and commutative quantales, or other similar axioms. The focus of this work is to study the different notion of 'completeness' a $\mathscr Q$-set may enjoy and their relations, completion functors, resulting reflective subcategories, their relations to relational morphisms. We establish the general equivalence of singleton complete $\mathscr Q$-sets with functional morphisms and the category of $\mathscr Q$-sets with relational morphisms; we provide two characterizations of singleton completeness in categorical terms; we show that the singleton complete categorical inclusion creates limits.