A category is called concrete if it is equipped with faithful functor to $\mathrm{Set}$. Let us say a category $C$ can be made concrete if there exists a faithful functor $F \colon C \to \mathrm{Set}$.

A famous result of Freyd, from his paper Homotopy is not concrete, gives an example of a category that cannot be made concrete. Let $\mathcal{T}$ be any category with

• some class of pointed topological spaces containing all finite-dimensional CW complexes as objects

and

• homotopy classes of basepoint-preserving continuous maps as morphisms.

Then Freyd showed $\mathcal{T}$ cannot be made concrete.

I believe one can deduce from this that the usual homotopy category of topological spaces, $\mathrm{Ho}(\mathrm{Top})$, cannot be made concrete.

More generally, given categories $C$ and $D$, write $C \le D$ if there exists a faithful functor from $C$ to $D$. This gives a preorder on categories which I'll call the concreteness preorder.

Freyd's result shows $\mathrm{Ho}(\mathrm{Top}) \nleq \mathrm{Set}$. On the other hand, $\mathrm{Set} \le \mathrm{Ho}(\mathrm{Top})$ thanks to the functor sending any set to the corresponding discrete space and any function to the corresponding equivalence class of maps.

So, we can say $\mathrm{Set} \lt \mathrm{Ho}(\mathrm{Top})$ in the concreteness preorder.

This is a rather simple-minded but precise way of saying that set theory can be embedded in homotopy theory but not conversely.

My question is simply, what other nice results are known about the concreteness preorder on categories?

For example are there some well-studied categories that are very high in this order? I imagine the homotopy category of $(\infty,n+1)$-categories is higher up than homotopy category of $(\infty,n)$-categories.