Concrete Category - Counter-examples

Counter-examples

The category hTop, where the objects are topological spaces and the morphisms are homotopy classes of continuous functions, is an example of a category that is not concretizable. While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. The fact that there does not exist any faithful functor from hTop to Set was first proven by Peter Freyd. In the same article, Freyd cites an earlier result that the category of "small categories and natural equivalence-classes of functors" also fails to be concretizable.