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This question is related to this one: Local smallness and (higher) topoi which has not yet been answered.

The $2$-category of topoi and geometric morphisms is not locally small. (As I mentioned in the question above, for example, if $A$ is the classifying topos for abelian groups, the category of geometric morphisms from $Set$ to $A$ is equivalent to the category of all abelian groups, which is not small.)

**Question:**Is the $2$-category of topoi and only etale geometric morphisms locally small?

Note: I'm actually interested in the corresponding statement for infinity topoi.

Remark: Notice that this $2$-category is locally a topos, so, it is locally locally small (sorry for the weird wording). That is, if $E$ is a topos, the slice 2-cat over $E$ consisting of etale morphisms over $E$ is known to be equivalent to $E$ itself (and hence is really a 1-category). There is a corresponding result for infinity topoi as well. Perhaps there is a way to use this to prove the result.

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  • $\begingroup$ Pardon for the comment, but to me it feels like there is some need for clarification: as stated, you already say that hom-categories are topoi; if you talk about infinity topoi, which are usually presumed to have all small (co)limits, that already says goodbye to smallness. What is meant precisely? $\endgroup$
    – Edouard
    Commented Nov 12, 2013 at 16:34
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    $\begingroup$ @Edouard: I think you misread what I wrote. I did not say that the Hom categories are topoi (they are certianly not!). What I said is, if I fix an object $E$ and look at the slice 2-category over $E,$ that is, the 2-category of etale geometric morphisms over $E,$ then this is equivalent to the underlying $1$-category of the topos $E.$ $\endgroup$ Commented Nov 12, 2013 at 17:30

1 Answer 1

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There is only a set of isomorphism classes of étale geometric morphisms between any two Grothendieck toposes. In fact, the same is true for essential geometric morphisms.

Recall that Grothendieck toposes are locally presentable categories, and that the left adjoint of a $\kappa$-accessible functor between $\kappa$-accessible categories must send $\kappa$-compact objects in the domain to $\kappa$-compact objects in the codomain. Since there is only a set of isomorphism classes of $\kappa$-compact objects in a $\kappa$-accessible category, we deduce that there is only a set of isomorphism classes of $\kappa$-accessible right adjoints between $\kappa$-accessible categories. But if $\mathcal{E}$ and $\mathcal{F}$ are locally $\kappa$-presentable toposes and $f : \mathcal{F} \to \mathcal{E}$ is an essential geometric morphism, then the inverse image functor $f^* : \mathcal{E} \to \mathcal{F}$ is a $\kappa$-accessible right adjoint. Thus there is only a set of isomorphism classes of essential geometric morphisms $\mathcal{F} \to \mathcal{E}$.

I imagine the same argument works verbatim for $(\infty, 1)$-toposes, but I do not know the details well enough to be sure.

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