Is there any category theoretic proof for independence of Continuum Hypothesis?

Question

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.

Answer 1

You can try the following book:

Saunders MacLane, Ieke Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer New York, 1994.

Answer 2

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]).