TITLE: Category theory $\subset$ Set theory or vice versa?
QUESTION [5 upvotes]: I just started reading the ABC of category theory using the appendix of a text, the first chapter of a text that I have never read, and above all (I found out now that they handle well the theory) the wikipedia pages. I want to know only this: in all three sources are given the definition of the category, and in all three I noticed that it's not possible to treat categories in set theory. The reason strikes the eye immediately when it is assumed that the objects are contained in classes, in particular the example of the category of sets. I ask then how it is treated the theory because from what I'm reading I do not understand (indeed in the second source it seems that develops internally to set theory). I think it needs some extension of set theory for treating them.

REPLY [4 votes]: Both category theory and set theory can be seen as formal theories in the general sense of mathematical logic. It is well-known that category theory can be developed inside set theory (perhaps including Grothendieck's axiom of universes). But Lawvere has also shown the converse, via the ETCS. More generally one can quote Grothendieck's topos theory, which is actually a categorical refinement of set theory (which is just the topos corresponding to a point). So in somse sense category theory and set theory contain each other.