Strongly finitary metric monads are too strong

Abstract: Varieties of quantitative algebras are fully described by their free-algebra monads on the category Met of metric spaces. For a longer time it has been an open problem whether the resulting enriched monads are precisely the strongly finitary ones (determined by their values on finite discrete spaces). We present a counter-example: the variety of algebras on two close binary operations yields a monad which is not strongly finitary. A full characterization of free-algebra monads of varieties is: they are the semi-strongly finitary monads, i.e., weighted colimits of strongly finitary monads (in the category of finitary monads).