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Discrete equational theories
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- Journal:
- Mathematical Structures in Computer Science / Volume 34 / Issue 2 / February 2024
- Published online by Cambridge University Press:
- 22 January 2024, pp. 147-160
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- Article
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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.
