Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
    
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

1 Answer 1

up vote 2 down vote accepted

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 $\mathcal{E}/E$.

Added later:

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.

share|improve this answer
    
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
1  
@ZhenLin: Indeed, but this related example will do - mathoverflow.net/questions/98729/… –  François G. Dorais Nov 11 '13 at 13:08

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.