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My doubt is simple. I have some possible foundations for category theory. If i'm doing category with NBG as a foundation, and if i define Cat as the category of all small categories i have in conclusion that Cat is a proper class. My questions is: In MacLane's book, he give a foundation to category with ZFC and assuming the existence of one universe $U$ (one set) where the elements of $U$ are called small, and he defines all the categories in terms of $U$, in particular i have $Cat_{small}$ the category of all small categories, but in this case, i'm not talking about the same thing that in NBG? Is $Cat$ a set in the definition of MacLane?

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No, $Cat$ is not a small category in Mac Lane's definition. The objects of $Cat$ are all of the elements in $U$, the universe. Thus $ob(Cat)=U$, which is a proper set. Every category though is required to be locally small, meaning that each hom set is actually a set, i.e., an element of $U$.

It should be noted that it is often convenient to have a larger hierarchy of universes. The logical treatment of that was done by Grothendieck, and, if I recall correctly, a proof that Grothendieck universes are consistent with ZF(C).

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    $\begingroup$ Well, $Cat_U$ (as one might call it) isn't small in Mac Lane's sense, but it is a set, I believe. $\endgroup$ May 26, 2014 at 11:35
  • $\begingroup$ I would not say that universes are consistent with ZFC – that is something we cannot possibly prove using ZF(C) alone. $\endgroup$
    – Zhen Lin
    May 26, 2014 at 12:46

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