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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.03180 |
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| _version_ | 1866908813941014528 |
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| author | Adamek, Jiri |
| author_facet | Adamek, Jiri |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_03180 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Strongly finitary metric monads are too strong Adamek, Jiri Category Theory 18A30, 18C20, 68N30 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. |
| title | Strongly finitary metric monads are too strong |
| topic | Category Theory 18A30, 18C20, 68N30 |
| url | https://arxiv.org/abs/2601.03180 |