Do equivalence relations form a monadic category? [closed]

Solution 1:

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.