Actually, it didn't resolve the crisis. In fact, categories, computability theory, and most mathematical findings of the last 150 years are a direct result of s

Actually, it didn't resolve the crisis. In fact, categories, computability theory, and most mathematical findings of the last 150 years are a direct result of stepping around the crisis, namely Russell's paradox. My point is not that category theory is more fundamental than set theory (though set theory can be described in terms of categories), but rather that it isn't derivative. You don't need the formalisms of set theory to have category theory.