# Are topoi and etale geometric morphisms locally small?

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.

• 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? Nov 12, 2013 at 16:34
• @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 $E.$ Nov 12, 2013 at 17:30

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.