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Is the category of equivalence relations and relation-preserving functions a monadic category (i.e. isomorphic to the Eilenberg-Moore category of some monad on ${\bf Set}$)?

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  • $\begingroup$ The standard term for what you are calling a "monadic category" is an algebraic category. It doesn't generally make sense to talk about "monadic categories". $\endgroup$
    – varkor
    Nov 26 '21 at 21:38
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    $\begingroup$ I'm using the term used by Adamek et al "The Joy of Cats" $\endgroup$ Nov 26 '21 at 21:45
  • $\begingroup$ They use the term for a concrete category, which strictly speaking is a faithful functor into $\mathbf{Set}$ (rather than the category that is its domain). Otherwise, it's not clear over which category you're monadic. It's a subtle distinction, but one it's worth being aware of. $\endgroup$
    – varkor
    Nov 26 '21 at 22:37
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It is rarely the case that "relational" structures (as opposed to "algebraic" structures) are monadic over $\textbf{Set}$. In this case there is an easy-to-see obstruction. Given any set $X$ with at least two elements, there are at least two different equivalence relations on $X$, which by analogy with topology we might call discrete and indiscrete. The identity map on $X$ is also a map from the discrete equivalence relation to the indiscrete equivalence relation. However, there is no inverse map from the indiscrete equivalence relation to the discrete equivalence relation. This shows that the forgetful functor to $\textbf{Set}$ is not conservative. Since monadic functors are conservative, we conclude it cannot be monadic.

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