Most of category theory as used in ordinary mathematics can be formulated in ZFC. The category of all groups, etc. is just a propositional formula. If you pre

Most of category theory as used in ordinary mathematics can be formulated in ZFC. The category of all groups, etc. is just a propositional formula. If you prefer to explicitly talk about classes, then you can go with NBG, which is a conservative extension of ZFC. Grothendieck needed an extra axiom because he wanted to consider large Grothendieck topologies. The consensus seems to be that universes are avoidable: https://mathoverflow.net/questions/35746/inaccessible-cardinals-and-andrew-wiless-proof https://mathoverflow.net/questions/35746/inaccessible-cardin...