Discrete equational theories

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Authors

ROSICKÝ Jiří

Year of publication 2024
Type Article in Periodical
Magazine / Source Mathematical Structures in Computer Science
MU Faculty or unit

Faculty of Science

Citation
Web https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/discrete-equational-theories/B68D91B64C2E6EC95C441A67CD9A24A4
Doi http://dx.doi.org/10.1017/S096012952400001X
Keywords Enriched equational theory; enriched monad; Birkhoff subcategory
Description On a locally $\lambda$-presentable symmetric monoidal closed category $\mathcal {V}$, $\lambda$-ary enriched equational theories correspond to enriched monads preserving $\lambda$-filtered colimits. We introduce discrete $\lambda$-ary enriched equational theories where operations are induced by those having discrete arities (equations are not required to have discrete arities) and show that they correspond to enriched monads preserving preserving $\lambda$-filtered colimits and surjections. Using it, we prove enriched Birkhof-type theorems for categories of algebras of discrete theories. This extends known results from metric spaces and posets to general symmetric monoidal closed categories.
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