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Working through Categories for the Working Mathematician, there was a point that has confused me. Throughout chapter 2, section 2, Mac Lane describes a (family of) functors which map locally small categories to Set by $b \mapsto \hom(a, b)$ and $f \mapsto f_*$. At the top of page 35 he then describes a method for constructing a similar functor on categories that aren't locally small:

To include categories [that aren't locally small], we can proceed as follows: Given a category $C$, take a set $V$ large enough to include all subsets of the set of arrows of $C$...

I'm confused by this: how could $V$ possibly be a set and not a proper class? Indeed, how could all the arrows of $C$ not be a proper class? Since $C$ is not locally small, there is some $a, b$ such that $\hom(a, b)$ is a proper class. Then, $\hom(a, b)$ is a subset of all arrows of $C$, which in turn has an obvious bijection to $V$, by sending a given arrow to its singleton. Thus, $\hom(a, b) \subseteq V$, so $V$ must also be a proper class. What am I missing?

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Mac Lane defines "small sets" in terms of universes. In particular, on page 12 he says:

We shall assume that there is a big enough set $U$, the "universe", then describe a set $x$ as "small" if it is a member of the universe.

In section I.6, he elaborates that by "big enough set" he actually means a Grothendieck universe. So for him, a "large" category is still a set, just one which is larger than the chosen universe. For categories that are actually proper classes, he uses the term "metacategory" instead (and does not consider them to be "categories").

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  • $\begingroup$ Very helpful thank you. I feel like this answer is a little unsatisfying though, because it doesn't answer the question: does such a construction exist for Mac Lane's metacategories? $\endgroup$ – Duncan Ramage Dec 15 '17 at 6:00
  • $\begingroup$ That depends what you mean by "exist". The hom operations will be definable, but they take values in proper classes, and there is no target (meta)category for them to be landing in. $\endgroup$ – Eric Wofsey Dec 15 '17 at 6:04
  • $\begingroup$ The lack of a target metacategory is probably what would make me say that they don't exist. Thanks. $\endgroup$ – Duncan Ramage Dec 15 '17 at 19:26

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