Is Rel a topos?

Question

Is the category Rel of sets and relations a topos?

I've done a few Google searches about this question but I haven't found any answers either way. And I can't recall any answers either way in any of the topos theory books that I've read.

Thanks!

Answer 1

The short answer is no.

• If by $\mathbf{Rel}$ you mean the category whose objects are sets and whose morphisms are relations, then $\mathbf{Rel}$ is not even cartesian closed: it has an object that is both initial and terminal, but not every object is such, so we may apply this argument.
• If by $\mathbf{Rel}$ you mean the category whose objects are sets equipped with a binary relation and whose morphisms are the maps preserving that binary relation, then $\mathbf{Rel}$ is not balanced, i.e. there is a non-isomorphism that is both a monomorphism and an epimorphism. But every topos is balanced.