# Is ${\bf Set}$ a Mal'cev category?

I read something on Mal'cev categories. It is well-known that the category of groups is an example of Mal'cev category. However, the category of sets is never given as an example of Mal'cev categories. I suppose it is not, but I have no counterexamples (or a proof if it is a Mal'cev category). Can you help me?

• Incidentally, $\textbf{Set}^\textrm{op}$ is a Mal'cev category. Mar 15, 2022 at 22:42

Just going from the $$n$$-lab's description of a Mal'cev category
it seems clear that $$\underline{Set}$$ cannot be a Mal'cev category. In $$\underline{Set}$$ an internal relation is just a relation, and reflexive and equivalence have their usual meanings. Since a set can have many reflexive relations that are not equivalences, $$\underline{Set}$$ cannot satisfy the condition.