TITLE: Is there any category theoretic proof for independence of Continuum Hypothesis?
QUESTION [6 upvotes]: Both of set theory and category theory could be a foundation for mathematics. Many set theoretic arguments could be translated to a category theoretic argument and vice versa. 

Question: Is there any category theoretic proof for independence of Continuum Hypothesis ($CH$)? What about more complicated arguments like independence of Martin's Axiom ($MA$)? Any reference for category theoretic interpretation of purely set theoretic arguments is also welcome.

REPLY [3 votes]: There is also this paper on arXiv:  "The logic of sheaves, sheaf forcing and the independence of the Continuum Hypothesis" by J. Benavides (arXiv: 1111.5854v1 [math.LO]).