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Following onto the previous question Are category of groups and abelian groups topoi? I find myself wondering: Are there any varieties of algebras, other than ones equivalent to $M{-}\mathsf{Set}$ for some monoid $M$, which are topoi? If not, is it possible that there are monads on $\mathsf{Set}$, other than ones isomorphic to $M \times {-}$, such that the Eilenberg-Moore category is a topos?

One trivial counterexample is the terminal category: this is a topos, and equivalent to a variety of algebras, yet for any monoid $M$ the category of $M$-sets contains at least two nonisomorphic objects, the initial object (with empty underlying set) and the terminal object (with terminal underlying set and trivial action of $M$).

(And if we relax the "topos" requirement to "pretopos", then for example the category of compact Hausdorff spaces is monadic over $\mathsf{Set}$.)

My question is: are there any other counterexamples?

Since any variety of algebras is complete and cocomplete, and is generated by $F(1)$ for $F$ the left adjoint to the forgetful functor $U$, that would be equivalent to being a Grothendieck topos. Going by the Giraud axioms in the linked question, I think a variety of algebras should in general satisfy all those axioms except for G3 and G5. So if this is correct, an equivalent question would be: are there any other varieties of algebras in which coproducts are disjoint and commute with pullbacks?

Daniel Schepler
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    I believe Johnstone [When is a variety a topos?] answers this question. (I do not have access to the article.) – Zhen Lin Mar 02 '25 at 04:27

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The varieties that are toposes are completely characterised in §4 of Garner's Cartesian closed varieties II: links to algebra and self-similarity (which clarifies aspects of Johnstone's earlier paper on the topic When is a variety a topos?, which Zhen Lin mentions in the comments). See Theorem 4.7 in particular.

For instance, the Jónsson-Tarski topos is an example of a variety that is a topos but is not a category of $M$-sets (since the forgetful functor to Set does not preserve binary coproducts): see §6 in Garner's paper.

varkor
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  • OK, thanks, the link does look interesting. Though the answer could be a bit more useful if it weren't essentially a link-only answer. Maybe for example you could extract the example of the category of Jónsson–Tarski algebras? (Though it might need a bit more work to show it's not an $M$-set category. If it were, $M$ would have to have underlying set of the reduced expressions in $\ell, r, m, x_0$ which isn't naturally a monoid using $m$, but that doesn't necessarily mean there isn't some other appropriate monoid structure on that set.) – Daniel Schepler Mar 02 '25 at 20:06
  • @DanielSchepler: I added a specific counterexample to my answer. – varkor Mar 02 '25 at 20:45