# Local smallness and (higher) topoi

The $2$-category of topoi and geometric morphisms is not locally small. 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. This problem of course persists for higher topoi, since ordinary topoi embedd into them.

Question: Suppose that $\mathcal{E}$ and $\mathcal{F}$ are two topoi (or higher topoi) such that I know that $Hom\left(\mathcal{E},\mathcal{F}\right)$ is small. It is easy to show that for any $F$ in $\mathcal{F},$ $Hom\left(\mathcal{E},\mathcal{F}/F\right)$ is also small. Can it also be shown that for all $E$ in $\mathcal{E},$ $Hom\left(\mathcal{E}/E,\mathcal{F}\right)$ is small?

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Interesting question. If you know the claim for a generating family of objects $E$, then it should be possible to deduce the claim for general $E$ using descent theory. –  Zhen Lin Oct 26 '13 at 18:48
Thanks for the comment Zhen. Yes, this occured to me, and I know how to prove it. However, this would not help me very much in the specific situation I am considering. –  David Carchedi Oct 26 '13 at 19:04

Consider a theory which has no models in $\mathrm{Set}$, but has a model in $\mathrm{Sh}(L)$ for some locale $L$. For example, the theory $\mathcal{CLF}$ of complete linearly ordered fields with more than $\sharp \mathbb{R}$ number of elements will do. For $L$ we can take Barr or Diaconescu covering of $\mathbb {B}\mathcal {CLF}$. Take some non-small theory, like abelian groups $\mathcal{A}b$, and consider a classifying topos $\mathrm {B} T$ for pairs $T = (K: \mathcal{CLF}, A: \mathcal{A}b)$. For the topos $\mathcal E$ consider $$\mathrm {Sh} (L + pt) = \mathrm {Sh}(L) + \mathrm {Set}$$ An etale map $L \to L + pt$ corresponds to some sheaf $E \in \mathcal E$, and $\mathcal {E}/E = \mathrm {Sh}(L)$.

$T$ will have no models in $\mathcal{E}$, but a non-small category of models in $\mathcal{E}/E$.

Instead of $\mathcal{CLF}$ we could take any theory without $\mathrm{Set}$-models. For example we could take a geometric theory, corresponding to some locale without points (just in case if there happen to be some obscure smallness problems with the following statements).

For any topos $\mathcal{F}$ we have $$Mod_T (\mathcal{F}) = Mod_{\mathcal{CLF}}(\mathcal{F}) \times Mod_{\mathcal{A}b} (\mathcal{F})$$ since a model of $T$ is just a pair of models ($Mod_T(\mathcal{F})$ is the category of T-models in $\mathcal{F}$). Since $L$ is the Barr covering for $\mathcal{CLF}$, $Mod_{\mathcal{CLF}}(\mathrm{Sh}(L))$ is a non-empty category. Thus to prove that $Mod_T(\mathcal{F})$ is non-small, we only need to prove it for $Mod_{\mathcal{A}b}(\mathcal{F})$. The category $Mod_{\mathcal{A}b}(\mathrm{Set})$ is clearly non-small. I claim that the category of constant $\mathcal{A}b$-valued sheaves on $\mathrm{Sh}(L)$ is a non-small subcategory of $Mod_{\mathcal{A}b}(\mathcal{F})$.

The terminal morphism $p\colon L \to pt$ gives an adjoint pair of functors $p_* \colon \mathrm{Sh}(L) \leftrightharpoons \mathrm{Set} \colon p^*$, $p^* \vdash p_*$. Sheaves on locale $L$ correspond to etale spaces over $L$, and the pullback functor $p^*$ is simply the pullback of corresponding etale spaces, i.e. it maps $S\in \mathrm{Set}$ to $\left( L\otimes S \to L \right) \in Et(L)$, the morphism to $L$ being the obvious "collapsing fibers" projection $L\otimes S \to L \otimes pt = L$. Here $L \otimes S$ stands for a coproduct of $S$ copies of $L$ in category of locales. Now the statement that $p^*: \mathcal{A}b(\mathrm{Set}) \to \mathcal{A}b(\mathrm{Sh}(L))$ is injective looks like something that should be obviously true (for $A\in \mathcal{A}b(\mathrm{Set})$ the sheaf of abelian groups $p^*(A)$ should have global sections roughly like $A^{\pi_0(L)}$), but at the moment I can only formalize the proof in the following roundabout way.

An isomorphism $p^*(A) \simeq p^*(B)$, $A,B \in \mathcal{A}b(\mathrm{Set})$ would give an isomorphism of etale spaces $L\otimes A \simeq L\otimes B$. Since $\mathcal{L}oc = \mathcal{F}rm^{op}$, this would give an isomorphism $\mathcal{O}(L)^A \simeq \mathcal{O}(L)^B$, $\mathcal{O}\colon \mathcal{L}oc^{op} \simeq \mathcal{F}rm$. The product in $\mathcal{F}rm$ is induced from the product in $\mathrm{Set}$, like in any algebraic theory. Thus if $|A| > |\mathcal{O}(L)|$ and $|B| > |\mathcal{O}(L)|$, then $\mathcal{O}(L)^A$ and $\mathcal{O}(L)^B$ are non-isomorphic even as sets, moreso as frames. Here $|A|$ stands for cardinality. Since we have a non-small set of groups with cardinality greater than $|\mathcal{O}(L)|$ (e.g. all big enough free groups), the statement is proved.

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I am not convinced... Why should there be model of CLF in some $Set/E$ ? This seem impossible to me : if $e \in E$ then there is a geometric morphism $e: Set \rightarrow Set/E$ and any model $F$ of CLF in $Set/E$ should give $e^* F$ in $Set$... –  Simon Henry Nov 11 '13 at 12:04
@Anton: Can you give me an example of such an $E$ where this works? –  David Carchedi Nov 11 '13 at 12:06
@AntonFetisov The general principle sounds good, but the specific example seems odd. Order-completeness is not a first-order property, even if you allow infinitary disjunctions. –  Zhen Lin Nov 11 '13 at 12:13
@Simon: What is the geometric morphism $e$ to which you are referring? The etale morphism (which is the unique map to the terminal topos in this case) goes the other way around. –  David Carchedi Nov 11 '13 at 12:25
@ZhenLin: Indeed, but this related example will do - mathoverflow.net/questions/98729/… –  François G. Dorais Nov 11 '13 at 13:08