In *The prospects of unlimited category theory: doing what remains to be done*, 2014 (The Review of Symbolic Logic, **8** (2015) pp 306-327, link), Ernst discusses Feferman's program, described in *Foundations of unlimited category theory: what remains to be done* (The Review of Symbolic Logic, **6** (2013) pp 6-15, link), for "unlimited category theory", characterized by three desirable axioms:

- There exists a category containing all objects of a given type (set, group, topological space, etc). It should literally include all such objects with no size limitations.
- For any two categories $A$, $B$, there should exist a functor category $B^A$.
- It should enable all standard constructions such as $\mathbb{N}$ and sums and products, etc.

Ernst shows that these axioms are inconsistent, by proving a version of Cantor's theorem for the category of reflexive graphs, thereby demonstrating a version of Cantor's paradox, showing that desired axioms are inconsistent.

He concludes that the search for such foundations is misguided, and so are objections to a ZFC-based set theoretic foundation for category theory.

To me this seems analogous to the situation with formal set theory with unrestricted comprehension in the days of Russell's paradox. Or the situation Girard's paradox for Martin-Löf type theory. Those formalisms were able to be saved. Is there any hope of salvaging Feferman's unlimited category theory?

What is the state of the art in category-theoretic foundations for category theory in 2018?

category-theoreticfoundations for category theory.) $\endgroup$ – Mike Shulman Jun 8 '18 at 13:56