Homotopical algebra is not concrete

Abstract

We generalize Freyd’s well-known result that “homotopy is not concrete”, offering a general method to show that under certain assumptions on a model category \(\mathcal {M}\), its homotopy category \(\textsc {ho}(\mathcal {M})\) cannot be concrete. This result is part of an attempt to understand more deeply the relation between set theory and abstract homotopy theory.