# ZF(C) and category theory

Is there an axiomatisation of some kind of category theory and a definition of sets in this framework such that the axioms of ZF resp. ZFC are theorems?

• If you formulate everything correctly, ECTS + separation is equivalent to ZFC. There is a translation you need to do because ECTS is two-sorted. See Tom Leinster's nice summary article: arxiv.org/abs/1212.6543 – Harry Gindi Jan 20 '19 at 12:33
• Thanks for the reference. – Jörg Neunhäuserer Jan 20 '19 at 12:48
• I think I meant replacement, not separation. It's whatever Leinster calls axiom 11. – Harry Gindi Jan 20 '19 at 13:19