I'm confused of background logic on category theory.
In ZFC set theory, we can construct new sets from existing sets by axioms, such as power set axiom, axiom of pairing etx.
I read first few pages of MacLane's category theory text and now I'm reading Tom Leinster's category theory text. Neither of these texts say whether we need some axioms or not, however, they are using some axioms in some sense without saying. I want to know what are the standard axioms for category theory.
Here are examples:
Firstly, how do we construct $A\times B$ where $A,B$ are categories? It is written in texts that "if we define $\operatorname{Obj}(A\times B)=\operatorname{Obj}(A)\times \operatorname{Obj}(B)$ and $\operatorname{Mor}((A_1,B_1),(A_2,B_2))=(\operatorname{Mor}(A_1,A_2),\operatorname{Mor}(B_1,B_2))$, then $A\times B$ forms a category". What kind of axiom would make this collecting possible?
Secondly, how do we construct a functor category $[A,B]$? How do we make "Collecting functors" process possible?
Thirdly, it is a theorem in text that "fully faith and essenially surjective functors are equivalences". However, to prove this, we need some kind of axiom of choice for category theory.
What would be the standard axioms?
Best Answer
The standard axioms vary: they're either ZFC with an axiom of choice for proper classes, some set theory such as NBG that axiomatizes classes more thoroughly, or ZFC with Grothendieck universes, so that "large" categories are interpreted as still being small, but relative to a larger "universe" of sets. There have been efforts to axiomatize category theory without set theory, most notably ETCC, the elementary theory of the category of categories, but these have not proven to be sufficient as a foundation.