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?
2026-04-09 09:08:14.1775725694
Category Cat and set theory
239 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
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).