Exercices de style: A homotopy theory for set theory

We construct a model category (in the sense of Quillen) for set theory, starting from two arbitrary, but natural, conventions. It is the simplest category satisfying our conventions and modelling the notions of finiteness, countability and infinite equi-cardinality. We argue that from the homotopy theoretic point of view our construction is essentially automatic following basic existing methods, and so is (almost all) the verification that the construction works. We use the posetal model category to introduce homotopy-theoretic intuitions to set theory. Our main observation is that the homotopy invariant version of cardinality is the covering number of Shelah’s PCF theory, and that other combinatorial objects, such as Shelah’s revised power function—the cardinal function featuring in Shelah’s revised GCH theorem—can be obtained using similar tools. We include a small “dictionary” for set theory in QtNaamen, hoping it will help in finding more meaningful homotopy-theoretic intuitions in set theory.