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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 ?

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  • $\begingroup$ 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. $\endgroup$ – Kevin Carlson Mar 7 '14 at 17:15
  • $\begingroup$ See this comment of t.b. $\endgroup$ – Zhen Lin Mar 7 '14 at 18:09
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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.

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