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Fix $\mathscr U$, $\mathscr V$ as two universes such that $\mathscr U\in\mathscr V$. Let $\mathsf{CAT}$ denote the category of categories $\mathcal C$ such that $\text{ob }\mathcal C\in\mathscr V$ and $\text{Mor }\mathcal C\in\mathscr V$ (that is, $\mathscr V$-small categories), and let $\mathsf{Cat}$ denote the category of categories $\mathcal C$ such that $\text{ob }\mathcal C\in\mathscr V$ and for each pair of objects $a$, $b$ in $\mathcal C$, $\hom(a,b)\in\mathscr U$(that is, $\mathscr V$-small and locally $\mathscr U$-small categories).

Given a category $\mathcal C\in\mathsf{Cat}$ and a subset $\Sigma$ of $\text{Mor }\mathcal C$. It's a well-known fact that the localization of $\mathcal C$ with respect to $\Sigma$ exists in $\mathsf{CAT}$, and we denote it by $\mathcal C[\Sigma^{-1}]$. My question is:

Suppose the localization of $\mathcal C$ with respect to $\Sigma$ exists in $\mathsf{Cat}$, can we deduce that $\mathcal C[\Sigma^{-1}]$ lies in $\mathsf{Cat}$ and is, consequently, the localization of $\mathcal C$ in $\mathsf{Cat}$?

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An answer with the opposite conclusion from my previous (incorrect, and now deleted) answer. Let's hope this is slightly more correct.

I assume that by "a subset $\Sigma$ of $\text{Mor } \mathcal{C}$" it's not intended that $\Sigma$ should be $\mathscr U$-small? Otherwise I think it is clear that $\mathcal{C}[\Sigma^{-1}]$ is $\mathscr U$-small.

Let $\{G_\alpha\}$ be a family of simple groups indexed by cardinals $\alpha$ in the universe $\mathscr U$, such that $G_\alpha\leq G_\beta$ when $\alpha\leq\beta$, and such that the cardinality of $G_\alpha$ is at least $\alpha$. For example, take $G_\alpha=\text{PSL}_2\left(\mathbb{Q}(X_\alpha)\right)$ where $\mathbb{Q}(X_\alpha)$ is the field of rational functions in a set $X_\alpha$ of cardinality $\alpha$, with the sets $X_\alpha$ nested.

Let $\mathcal{C}$ be the category with one object $c_\alpha$ for each cardinal $\alpha$, where $\hom(c_\alpha,c_\beta)=G_\beta$ for $\alpha\leq\beta$ and $\hom(c_\alpha,c_\beta)=\emptyset$ for $\alpha>\beta$, with the obvious composition given by multiplication in the groups $G_\alpha$.

Let $\Sigma$ be the collection of all morphisms.

Then if $F:\mathcal{C}\to\mathcal{D}$ is a functor that inverts $\Sigma$, an object $d$ in the image has compatible actions of $G_\alpha$ for every $\alpha$. If $\mathcal{D}$ is locally $\mathscr U$-small, this means the action of $G_\alpha$ must be trivial for $\alpha$ greater than the cardinality of $\hom(d,d)$, since $G_\alpha$ is simple, and hence for all $\alpha$.

So the localization in $\mathsf{Cat}$ exists, being the category with precisely one morphism $c_\alpha\to c_\beta$ for every $\alpha,\beta$.

But in the localization in $\mathsf{CAT}$, the set of endomorphisms of each object is $\bigcup_\alpha G_\alpha$.

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  • $\begingroup$ Thank you, very illustrative. $\endgroup$ – Censi LI Apr 12 '17 at 13:05

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