Varieties of quantitative algebras as categories

Classical varieties were characterized by Lawvere as the categories with effective congruences and a varietal generator: an abstractly finite regular generator which is regularly projective (its hom-functor preserves regular epimorphisms). We characterize varieties of quantitative algebras of Mardare, Panangaden and Plotkin analogously as metric-enriched categories. We introduce the concept of a subcongruence (a metric-enriched analogue of a congruence) and the corresponding subregular epimorphisms obtained via colimits of subcongruences. Varieties of quantitative algebras are precisely the metric-enriched categories with effective subcongruences and a subvarietal generator: an abstractly finite subregular generator which is subregularly projective (its hom-functor preserves subregular epimorphisms).