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There is a lot of an interaction between category theory and set theory. In 1965, one year after Cohen's proof of the independence of the continuum hypothesis, Vopenka gave a proof using sheaf theory, see here.

This is nowadays systematized in the topos theoretic interpretation of set theory, for which you should look up MacLane/Moerdijk, as Dylan Wilson pointed out. The authors give a proof of the independence of the continuum hypothesis.

Marta Bunge has given a topos theoretic proof of the independence of the Suslin hypothesis from ZFC in: Marta Bunge, Topos Theory and Souslin's Hypothesis. J.Pure & Applied Algebra 4 (1974) 159-187.

As for the reformulation of Vopenka's principle: It is equivalent to the statement that a locally presentable category can not have a large full discrete subcategory. This, and more, is nicely explained in Adamek/Rosicky's Locally Presentable and Accessible Categories

On the other hand I know of no worked out connection between homotopy theory and set theory, just the indirect one via type theory mentioned by Andrej.

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There is a lot of interaction between category theory and set theory. In 1965, one year after Cohen's proof of the independence of the continuum hypothesis, Vopenka gave a proof using sheaf theory, see here.

This is nowadays systematized in the topos theoretic interpretation of set theory, for which you should look up MacLane/Moerdijk, as Dylan Wilson pointed out. The authors give a proof of the independence of the continuum hypothesis.

Marta Bunge has given a topos theoretic proof of the independence of the Suslin hypothesis from ZFC in: Marta Bunge, Topos Theory and Souslin's Hypothesis. J.Pure & Applied Algebra 4 (1974) 159-187.

As for the reformulation of Vopenka's principle: It is equivalent to the statement that a locally presentable category can not have a large full discrete subcategory. This, and more, is nicely explained in Adamek/Rosicky's Locally Presentable and Accessible Categories

On the other hand I know of no worked out connection between homotopy theory and set theory, just the indirect one via type theory mentioned by Andrej.