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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.

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Vopenka's principle has a well-known category theoretic interpretation. –  François G. Dorais Aug 26 '13 at 3:46
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The interplay of (significantly) large cardinals and category theory has been the focus of recent research of Bagaria and his collaborators. From what I understand (from discussions with him), there were quite a few results in both fields that only needed researchers fluid enough in both ways of thinking to make concrete. For applications of supercompactness, see his paper with Carles Casacuberta and Adrian Mathias on Epireflections and supercompact cardinals, available at his webpage ay ICREA. –  Andres Caicedo Aug 26 '13 at 4:01
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The paper also presents generous references on VopÄ›nka's principle from a category theoretic point of view. See also their paper with Jiri Rosicky, Definable orthogonality classes in accessible categories are small, available at the ArxiV. –  Andres Caicedo Aug 26 '13 at 4:05
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Regarding question 1, Dustin Mulcahey and I are working on an article exploring various senses in which one has or does not have an analogue of the Kunen inconsistency in category theory. But this is not yet ready... –  Joel David Hamkins Aug 26 '13 at 4:34
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Here's an old one: in his paper "Adequate subcategories" (Illinois Journal of Mathematics 4 (1960), 541-552, Theorem 2.5), John Isbell showed that the full subcategory of Set consisting of a single countable set is codense if and only if there are no measurable cardinals. –  Tom Leinster Aug 26 '13 at 11:49

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