# categorical characterization of large cardinals

Question 1. Is there a categorical representation of Kunen's inconsistency result?

Question 2. Is there a categorical characterization of very large cardinals (in particular for strong and supercompact cardinals)?

Question 3. Is there a categorical characterization of $0^{\sharp}$?

Remark 1. A categorical characterization of other large cardinals is also welcome.

Remark 2. Andreas Blass in the paper "Exact functors and measurable cardinals" has proved that the existence of a measurable cardinal is equivalent to the existence of a non-trivial exact functor from the category of sets to the category of sets.

Would you please give references for such matters.

Remark 3. The following papers may have some information about the relation between category theory and large cardinals: