TITLE: Category and homotopy theoretic methods in set theory
QUESTION [8 upvotes]: I am looking for applications of category theory and homotopy theory in set theory and particularly in cardinal arithmetics. "Applications" in the broad sense of the word --- this would include theorems, definitions, questions, points of view (and papers) in set theory that could be motivated or understood with help of category theory and homotopy theory. I am aware of some applications of set theory in category theory, e.g. large cardinal axioms (Vopenka principle) are used to construct localisations in homotopy theory, but this is not what I am asking for. However, I would be interested to hear if Vopenka principle is equivalent to a statement in category or homotopy theory.
The reason for the question is that I am trying to better understand this sketch of an attempt to understand an invariant in PCF theory in terms of homotopy theory. I am most interested in applications to cardinal arithmetic.

REPLY [5 votes]: Peter Freyd wrote a paper, "The Axiom of Choice," in which he used topos-theoretic methods to prove that the axiom of choice is independent of (classical) Zermelo-Fraenkel set theory.  Unlike earlier topos-versions of set-theoretic independence proofs, Freyd's construction does not merely provide a category-theoretic view of a model that had already been considered by set theorists.  His models can be obtained by set-theoretic forcing methods (by another result of Freyd, in "All topoi are localic"), but those particular forcing constructions had not been considered until Andre Scedrov and I analyzed them (in "Freyd's models for the independence of the axiom of choice").  So I think it's fair to say that these models were a contribution from category theory to set theory.