Quantitative Algebras and a Classification of Metric Monads
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2022
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| author | Adámek, J. Dostál, M. Velebil, J. |
| author_facet | Adámek, J. Dostál, M. Velebil, J. |
| contents | Quantitative algebras are $Σ$-algebras acting on metric spaces, where operations are nonexpanding. Mardare, Panangaden and Plotkin introduced 1-basic varieties as categories of quantitative algebras presented by quantitative equations. We prove that for the category $\mathsf{UMet}$ of ultrametric spaces such varieties bijectively correspond to strongly finitary monads on $\mathsf{UMet}$. The same holds for the category $\mathsf{Met}$ of metric spaces, provided that strongly finitary endofunctors are closed under composition.
For uncountable cardinals $λ$ there is an analogous bijection between varieties of $λ$-ary quantitative algebras and monads that are strongly $λ$-accessible. Moreover, we present a bijective correspondence between $λ$-basic varieties as introduced by Mardare et al and enriched, surjections-preserving $λ$-accesible monads on $\mathsf{Met}$. Finally, for general enriched $λ$-accessible monads on $\mathsf{Met}$ a bijective correspondence to generalized varieties is presented. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_01565 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quantitative Algebras and a Classification of Metric Monads Adámek, J. Dostál, M. Velebil, J. Category Theory Quantitative algebras are $Σ$-algebras acting on metric spaces, where operations are nonexpanding. Mardare, Panangaden and Plotkin introduced 1-basic varieties as categories of quantitative algebras presented by quantitative equations. We prove that for the category $\mathsf{UMet}$ of ultrametric spaces such varieties bijectively correspond to strongly finitary monads on $\mathsf{UMet}$. The same holds for the category $\mathsf{Met}$ of metric spaces, provided that strongly finitary endofunctors are closed under composition. For uncountable cardinals $λ$ there is an analogous bijection between varieties of $λ$-ary quantitative algebras and monads that are strongly $λ$-accessible. Moreover, we present a bijective correspondence between $λ$-basic varieties as introduced by Mardare et al and enriched, surjections-preserving $λ$-accesible monads on $\mathsf{Met}$. Finally, for general enriched $λ$-accessible monads on $\mathsf{Met}$ a bijective correspondence to generalized varieties is presented. |
| title | Quantitative Algebras and a Classification of Metric Monads |
| topic | Category Theory |
| url | https://arxiv.org/abs/2210.01565 |