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?
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asked Mar 15, 2022 at 22:08
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2$\begingroup$ Incidentally, $\textbf{Set}^\textrm{op}$ is a Mal'cev category. $\endgroup$– Zhen LinMar 15, 2022 at 22:42
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Just going from the $n$-lab's description of a Mal'cev category
A Malcev category is a left exact category (= having finite limits) in which any reflexive internal relation is an equivalence relation.
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.
answered Mar 15, 2022 at 22:16