Is it possible to construct ZFC set theory inside category theory?

It's entirely possible I don't understand what I am talking about, but I know that ZFC stands as a good foundation for much of mathematics and that category theory stands as a good foundation for other areas. Apparently TG set theory allows one to construct category theory inside of set theory, but is the reverse true? Is it possible to use category theory to construct set theory?

As in, is there an axiomatic version of category theory that can not only address homological algebra and so on, but at the same time neatly contain ZFC's axioms as theorems?

• What does it even mean "use category theory to construct ZFC"? Feb 28, 2015 at 11:56
• Well, one can construct objects similar to natural numbers which adhere to Peano's axioms in ZFC. I'm asking if there is some analogue between category theory and ZFC, so if objects similar to sets which adhere to ZFC axioms can be constructed. Feb 28, 2015 at 12:12
• For one thing, there is no "canonical model" for $\sf ZFC$ to begin with. So maybe first you need to ask yourself what is the analogue of $\Bbb N$ in this context. Before you do ask this question on the site, be sure to look around in the foundations questions, since I'm pretty sure that I have answered that question (about "the intended interpretation for $\sf ZFC$") at least once or twice before. Feb 28, 2015 at 12:14
• @Asaf Karaglia Just found a book which purports to do exactly what I was asking - haven't read it yet, but I just thought I'd say that the idea has at least been thought up before. May 10, 2015 at 13:30
• Great. Let me point out, again, that $\sf ZFC$ has no "canonical" model, like $\Bbb N$ for $\sf PA$. Sure it's possible to do set theory over a categorical foundation, but there's still no obvious and canonical model of $\sf ZFC$. May 10, 2015 at 16:35