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arXiv · 2601.03180

Strongly finitary metric monads are too strong

Jiri Adamek
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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.

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Jiri Adamek. 2026-01-06. Strongly finitary metric monads are too strong. https://arxiv.org/abs/2601.03180

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