Strongly finitary metric monads are too strong

TL;DR

This paper investigates the nature of monads in metric algebra varieties, providing a counter-example to the assumption that strongly finitary monads fully characterize these varieties, and offers a new characterization of free-algebra monads.

Contribution

It demonstrates that strongly finitary monads are not sufficient to describe all varieties of quantitative algebras and characterizes free-algebra monads as 1-basic monads, expanding understanding of their structure.

Findings

Counter-example of a non-strongly finitary monad from a binary operation variety

Strongly finitary endofunctors are not closed under composition

Full characterization of free-algebra monads as 1-basic monads

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 1-basic monads, i.e., weighted colimits of strongly finitary monads (in the category of enriched finitary monads). As a consequence, strongly finitary endofunctors on Met are not closed under composition.