Quantitative Algebras and a Classification of Metric Monads
J. Adámek, M. Dostál, J. Velebil
TL;DR
The paper establishes a deep correspondence between classes of quantitative algebras (varieties) and monads on metric categories, identifying strongly finitary (and λ-ary) monads with the free-algebra monads of appropriate varieties. It develops enriched Kan-extension and foliation techniques to prove these equivalences for both $\,\mathsf{UMet}$ and $\,\mathsf{Met}$ (under compositionality assumptions), and extends the theory to complete ultrametric spaces and generalized/λ-basic settings. The work unifies finite and infinitary, as well as ultrametric and metric, frameworks, providing a Dubuc-style bridge between algebraic theories and monadic semantics in enriched metric contexts. It also clarifies the limits of ω-ary results and outlines a generalized, λ-parameterized landscape for presentable theories in metric categories.
Abstract
Quantitative algebras are $Σ$-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 $λ$ there is an analogous bijection between varieties of $λ$-ary quantitative algebras and monads that are strongly $λ$-accessible. Moreover, we present a bijective correspondence between $λ$-basic varieties as introduced by Mardare et al and enriched, surjections-preserving $λ$-accesible monads on $\mathsf{Met}$. Finally, for general enriched $λ$-accessible monads on $\mathsf{Met}$ a bijective correspondence to generalized varieties is presented.