This post is an exposition of the material in the paper “Homotopy is not concrete” by P. Freyd, of whose existence I learned from this MO discussion.

A category {\mathcal{C}} is concrete if there is a faithful functor {F: \mathcal{C} \rightarrow \mathbf{Sets}}. Most of the categories one initially encounters are in fact concrete: categories of groups, rings, modules, Lie algebras, and so on, and one can think of them as consisting of “structured sets” and “morphisms respecting that structure.” Every small category is concrete, because one can take the Yoneda embedding

\displaystyle \mathcal{C} \rightarrow \mathbf{Sets}^{\mathcal{C}^{op}}

followed by the product functor {\mathbf{Sets}^{\mathcal{C}^{op}}\rightarrow \mathbf{Sets}}.

Nonetheless, not every category is concrete, and the following example shows that a very natural one is not:


Theorem 1 (Freyd) The homotopy category {\mathcal{H}ot_*} of pointed spaces is not concrete.


In other words, a homotopy type is somehow too complex to be encoded simply as a set with appropriate structure.

The idea of the proof is essentially the following. In a category of structured sets, a given object can only have so many subobjects, because a set has only so many subsets. But there are categories where an object may have an enormous collection of subobjects, because the definition of a subobject is purely arrow-theoretic. So a category where objects can have lots of subobjects is probably not concrete. (more…)