Reference : http://arxiv.org/abs/0810.1279
For 2 days, I have searched what possible foundations for category theory are there and I think the reference link explains possible foundations nicely. The paper shows that each foundation has advantage and disadvantage over others, but the paper says nothing about disadvantage of Inaccessible cardinal foundation, so I guess it is the best one.
What is the standard set-theoretic foundation for Category theory? Inaccessible cardinal approach?
If so, when we say "set", do we mean only "small sets" in the language of category theory? That is, are we dropping off "large sets" and consider only "small sets"?