Quantaloid-enriched categories: Factorization, weak classifiers, and symmetry
Javier Gutiérrez García, Ulrich Höhle
TL;DR
The paper develops a unifying treatment of categories enriched over quantaloids, introducing new structural results and applying them to quantale-valued contexts. It establishes a factorization system (epi, extremal mono) for the category of quantaloid-enriched categories with left adjoint distributors, and proves that separated cocomplete $\mathcal{Q}$-categories support a weak subobject classifier under stability of $\mathcal{Q}$. It further analyzes the Cauchy completion and symmetry preservation, providing complete characterizations and illustrating with involutive quantales and quantale-valued sets; this yields a broad generalization of $\Omega$-valued/set-valued semantics and connects to topos-like structures in specialized cases. Overall, the work offers a coherent framework linking presheaves, cocompletion, symmetry, and quantale-valued logic within quantaloid-enriched categories, with concrete implications for quantale-valued sets and related semantic theories.
Abstract
This paper provides a comprehensive overview of some of the foundational properties of categories enriched over quantaloids, along with several new results. We demonstrate that the category whose objects are quantaloid-enriched categories and whose morphisms are left adjoint distributors admits an (epi, extremal mono)--factorization system. Furthermore, we prove that the category of cocomplete quantaloid-enriched categories satisfies the weak subobject classifier axiom, under stability conditions on the underlying quantaloid. As an application, we discuss how these structural results extend to quantale-valued sets, thereby generalizing the classical theory of $Ω$-valued sets.