Varieties of Quantitative or Continuous Algebras (Extended 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 ca...