Quantitative Algebras and a Classification of Metric Monads

TL;DR

This paper establishes a correspondence between categories of quantitative algebras and certain classes of monads on metric and ultrametric spaces, extending to uncountable cardinals and generalized varieties.

Contribution

It proves bijections between varieties of quantitative algebras and strongly finitary or accessible monads on metric and ultrametric spaces, generalizing previous classifications.

Findings

Bijective correspondence between varieties of quantitative algebras and strongly finitary monads on ultrametric spaces.

Extension of the correspondence to metric spaces with closure under composition.

Generalization to uncountable cardinals and enriched, surjections-preserving 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.