Varieties of Quantitative or Continuous Algebras (Extended Abstract)

TL;DR

This paper establishes a correspondence between varieties of quantitative and continuous algebras and strongly finitary monads on their respective categories, providing new insights into their structural properties and categorical characterizations.

Contribution

It proves bijective correspondences between varieties of enriched algebras and strongly finitary monads across multiple categories, extending the theoretical framework of algebraic structures in metric and domain theory.

Findings

Varieties of quantitative algebras correspond to strongly finitary monads on Met.

Varieties of continuous algebras correspond to strongly finitary monads on CPO.

Directed colimits commute with finite products in all cartesian closed categories.

Abstract

Quantitative algebras are algebras enriched in the category $\mathsf{Met}$ of metric spaces so that all operations are nonexpanding. Mardare, Plotkin and Panangaden introduced varieties (aka $1$-basic varieties) as classes of quantitative algebras presented by quantitative equations. We prove that they bijectively correspond to strongly finitary monads $T$ on $\mathsf{Met}$. This means that $T$ is the Kan extension of its restriction to finite discrete spaces. An analogous result holds in the category $\mathsf{CMet}$ of complete metric spaces. Analogously, continuous algebras are algebras enriched in $\mathsf{CPO}$, the category of $\omega$-cpos, so that all operations are continuous. We introduce equations between extended terms, and prove that varieties (classes presented by such equations) correspond bijectively to strongly finitary monads $T$ on $\mathsf{CPO}$. This means that $T$ is the Kan extension of its restriction to finite discrete cpos. (The two results have substantially different proofs.) An analogous result is also presented for monads on $\mathsf{DCPO}$. We also characterize strong finitarity in all the categories above by preservations of certain weighted colimits. As a byproduct we prove that directed colimits commute with finite products in all cartesian closed categories.