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.

fullembeddability. Keyword "universal categories". $\endgroup$ – Tim Campion Dec 25 '18 at 21:21