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?

  • 2
    $\begingroup$ Incidentally, $\textbf{Set}^\textrm{op}$ is a Mal'cev category. $\endgroup$
    – Zhen Lin
    Commented Mar 15, 2022 at 22:42

1 Answer 1


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.


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