Strongly finitary metric monads are too strong
Jiri Adamek
TL;DR
The paper addresses the problem of characterizing free-algebra monads on the category $ extbf{Met}$ of metric spaces, showing that strongly finitary monads do not exhaust these free-algebra monads. It introduces $1$-basic monads as weighted colimits of strongly finitary monads and proves that free-algebra monads of quantitative varieties are exactly the $1$-basic ones, linking varieties to enriched monads via a full, faithful correspondence. A central result is a counterexample: a variety defined by two $oldsymbol{ extε-close}$ binary operations whose free-algebra monad is not strongly finitary, demonstrating that strong finitarity is not preserved under composition. The paper further establishes that, while all unary-variant monads are strongly finitary, the general theory requires the $1$-basic framework to capture free-algebra monads, and it proves completeness results for the category of varieties and its relation with enriched monads. Overall, the work clarifies the structural landscape of quantitative algebras and provides a precise criterion (being $1$-basic) for when a monad arises from a variety.
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.