Exercices de style: a homotopy theory for set theory, I
  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. In a subsequent paper \cite{GaHa1} we give a homotopy theoretic dictionary of set theoretic concepts, most notably Shelah's covering number $\cov(\lambda, \aleph_1,\aleph_1,2)$, recovered from this model category.   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. 
Assaf Hasson, Misha Gavrilovich
math.LO