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.