Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have read on nlab and other sources that I can't backtrack that the localization process of a category can lead to size issue. More especially, starting from a locally small category $\mathsf C$ and $\mathcal W \subseteq \mathrm{Mor}\, \mathsf C$, the category $\mathsf C [\mathcal W^{-1}]$ might not be locally small. As intuitive as it is, I had some hard time finding such a size changing example.

The simplest I can find (if I do not mistake) is the following. For the sake of rigour, let say we work with Grothendieck universes (but I think it is exactly the same to work with a fixed model of some set theory [like ZFC] and proper classes). Fix Grothendieck universes $U$ and $V$ with $U \in V$, and let $S$ be $V$-small but not $U$-small (i.e. $S \in V$ but $S \notin U$). Then construct the category $\mathsf C$

  • whose objects are : $x_0$, $x_1$ and all $s \in S$ ;
  • whose morphisms are : $x_0 \to s$ for all $s \in S$, and $x_1 \to s$ for all $s \in S$, and of course the identity morphisms.

$$ \mathsf C : \qquad \begin{matrix} && \vdots &&\\ & \nearrow & s & \nwarrow &\\ x_0 & \rightarrow & \vdots & \leftarrow & x_1 \\ & \searrow & s' & \swarrow &\\ && \vdots & \end{matrix} $$

Then $\mathsf C$ is clearly $U$-locally small (the hom-sets are empty or singleton, so $U$-small). Then choose $\mathcal W = \{x_0 \to s : s \in S \}$ and localize. We end up with a category

$$ \mathsf C[\mathcal W^{-1}] : \qquad \begin{matrix} && \vdots &&\\ & \stackrel \swarrow \nearrow & s & \nwarrow &\\ x_0 & \leftrightarrows & \vdots & \leftarrow & x_1 \\ & \stackrel \nwarrow \searrow & s' & \swarrow &\\ && \vdots & \end{matrix} $$

which isn't $U$-locally small as $\hom_{\mathsf C[\mathcal W^{-1}]}(x_1,x_0) \simeq S \notin U$.

However, this example seems very artificial and ad hoc. What are the natural examples of size changing localization ?

share|improve this question
    
I don't know any such natural examples-rather this issue is just motivation to have a different model of your localization in specific cases such as the derived category of an abelian category. Nice example, anyway. –  Kevin Carlson Mar 7 at 17:15
    
See this comment of t.b. –  Zhen Lin Mar 7 at 18:09

1 Answer 1

up vote 2 down vote accepted

Here is a non-contrived example. Let $\mathbf{Site}$ be the category whose objects are small Grothendieck sites and whose morphisms are isomorphism classes of morphisms of sites. (A morphism $(\mathcal{C}, J) \to (\mathcal{D}, K)$ is a functor $\mathcal{D} \to \mathcal{C}$ that sends $K$-covering families to $J$-covering families.) Let $\mathbf{Topos}$ be the category of Grothendieck toposes and isomorphism classes of geometric morphisms. There is then a functor $\mathbf{Sh} : \mathbf{Site} \to \mathbf{Topos}$ that sends a site $(\mathcal{C}, J)$ to the sheaf topos $\mathbf{Sh} (\mathcal{C}, J)$. It is known that every geometric morphism between two Grothendieck toposes comes from a morphism of sites, so the functor $\mathbf{Sh}$ is essentially surjective on morphisms.

Let us say that a morphism of sites is a Morita equivalence if the functor $\mathbf{Sh}$ inverts it. I claim the localisation of $\mathbf{Site}$ with respect to the class $\mathcal{W}$ of Morita equivalences is not locally small. Certainly there is a comparison functor $\mathbf{Site} [\mathcal{W}^{-1}] \to \mathbf{Topos}$, by the universal property of localisation, and it must be essentially surjective on morphisms because $\mathbf{Sh}$ is. Moreover, $\mathbf{Site} [\mathcal{W}^{-1}] \to \mathbf{Topos}$ is full because any two sites for the same topos are Morita equivalent (by definition). But $\mathbf{Topos}$ is not locally small, so $\mathbf{Site} [\mathcal{W}^{-1}]$ cannot be locally small.

share|improve this answer

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.