Quantitative Algebras and a Classification of Metric Monads

Abstract

Quantitative algebras are $\Sigma$-algebras acting on metric spaces, where operations are nonexpanding. Mardare, Panangaden and Plotkin introduced 1-basic varieties as categories of quantitative algebras presented by quantitative equations. We prove that for the category $\mathsf{UMet}$ of ultrametric spaces such varieties bijectively correspond to strongly finitary monads on $\mathsf{UMet}$. The same holds for the category $\mathsf{Met}$ of metric spaces, provided that strongly finitary endofunctors are closed under composition. For uncountable cardinals $\lambda$ there is an analogous bijection between varieties of $\lambda$-ary quantitative algebras and monads that are strongly $\lambda$-accessible. Moreover, we present a bijective correspondence between $\lambda$-basic varieties as introduced by Mardare et al and enriched, surjections-preserving $\lambda$-accesible monads on $\mathsf{Met}$. Finally, for general enriched $\lambda$-accessible monads on $\mathsf{Met}$ a bijective correspondence to generalized varieties is presented.