Varieties of Quantitative Algebras and Their Monads

Quantitative Σ-algebras, where Σ is a signature with countable arities, are Σ-algebras equipped with a metric making all operations nonexpanding. They have been studied by Mardare, Panangaden and Plotkin who also introduced c-basic quantitative equations for regular cardinals c. Categories of quantitative algebras that can be presented by such equations for c = ℵ1 are called ω1-varieties. We prove that they are precisely the monadic categories , where is a countably basic monad on the category of metric spaces