Has category theory solved major math problems?
Solution 1:
Emily Riehl's wonderful book Category Theory in Context is a book-length answer to your question. You can skip straight to the epilogue at the very end.
But, in particular, I think that the proofs of the Weil conjectures were driven by the category theoretical structure that Grothendieck and his colleagues worked out. This included a proof of a version of the Riemann hypothesis that is relevant for number theory.