**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:

1) Adequate subcategories-Isbell

2) Small adequate subcategories-Isbell,

3) Structure of categories-Isbell,

4) Exact functors and measurable cardinals-Blass,

5) Exact functors, local connectedness and measurable cardinals-Adelman & Blass.

Epireflections and supercompact cardinals, available at his webpage ay ICREA. – Andrés Caicedo Aug 26 '13 at 4:01Definable orthogonality classes in accessible categories are small, available at the ArxiV. – Andrés Caicedo Aug 26 '13 at 4:05