Quantaloid-enriched categories: Factorization, weak classifiers, and symmetry

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 $\Omega$-valued sets.