assuming grothendieck universes, is it possible to have a small category equivalent the category of all sets?
mac lane in categories for the working mathematician assumes the existence of a grothendieck universe set, which forms
a category $\mathbf {Set}$ of all small sets. skimming the book, mac lane never states this, but he seems do assume or suggest that the category of sets is somehow equivalent to $\mathbf {Set}$. but mac lane here only speaks of metacategories as some sort of pre-foundational mathematical objects (for instance he has a metacategory of all classes), so maybe one has to be more precise here, so:
taking as a basis an axiomatic set theory with a solid notion of classes such as neumann-gödel-bernays or morse-kelley together with the existence of a (sufficiently large) grothendieck universe set, is there a small category equivalent to the category of all sets?
Best Answer
No. If $C$ and $D$ are equivalent categories, then they must have the same "cardinality" of isomorphism types (ie they must either both have set-many isomorphism types, and then must have the same cardinality of these, or both have proper-class-many isomorphism types). One way of seeing this is to note that, assuming global choice, two categories are equivalent if and only if they have isomorphic skeletons. Since $\mathbf{Set}$ has proper-class-many isomorphism types (one for each cardinal), it thus cannot be equivalent to a small category.