There is this old result by Freyd that "homotopy is not concrete": 

> Freyd, Peter. "[*Homotopy is not concrete.*](http://www.tac.mta.ca/tac/reprints/articles/6/tr6abs.html)" The Steenrod Algebra and Its Applications: A Conference to Celebrate NE Steenrod's Sixtieth Birthday. Springer Berlin Heidelberg, 1970.

In 2017 we say that 

> * there is a [homotopical category](https://ncatlab.org/nlab/show/homotopical+category) $(\bf Top, \cal W)$, such that the [localization](https://ncatlab.org/nlab/show/localization) ${\bf Top}\!\!\left[{\cal W}^{-1}\right]$ is not ($\bf Set$-)[concrete](https://ncatlab.org/nlab/show/concrete+category).

The key result here is the following lemma, called "Isbell condition":

Let ${\cal A}/A$ the class of arrows with codomain $A$ in a category $\cal A$, and for $f\in {\cal A}/A$ let $C(f,B)$ be the class of pairs $u,v \in A/{\cal A}/B$ such that $uf=vf$ as morphisms $\text{src}(f) \to A \to B$.

Define an equivalence relation $\asymp$ on ${\cal A}/A$ that says $f\asymp g$ iff $C(f,B)=C(g,B)$ for every $B\in\cal A$, and let $\textsf{S}({\cal A}/A)$ the quotient $\left({\cal A}/A\right)_{/\asymp}$.

> **Isbell condition**: $\cal A$ is concrete if and only if $\textsf{S}({\cal A}/A)$ is a set for every $A\in\cal A$.

If you follow the development of this, you find another, slightly older paper

> Freyd, Peter. "*On the concreteness of certain categories.*" Symposia Mathematica. Vol. 4. 1969.

that develops quite a bit this technology, and contains (Thm 4.1) a result that in 2017 we write as

> * the localization ${\bf Cat}\!\!\left[\text{Eqv}^{-1}\right]$ of the category of categories to equivalences (i.e. the homotopy category [$\textsf{Ho}({\bf Cat}_\text{folk})$](https://ncatlab.org/nlab/show/canonical+model+structure+on+Cat) is not concrete.

Now, following [Christie's meta-theorem](http://community.agathachristie.com/discussion/706/which-novel-quote) it's easy to wonder if there is a pattern, and maybe a proof, here.

Freyd's theorem is as old as Quillen's definition of a model category, so I doubt that Freyd ignored that you can ask the following question:

> * Let $\mho$ be a universe. If $\cal M \in \mho\text{-}{\bf Cat}$ has a model structure, how often is the localization $\textsf{Ho}(\cal M)$ a ($\mho\text{-}\bf Set$-)concrete category?

(One could argue that this result really belongs to the world of homotopical categories and should be stated therein: it should, but a model structure is highly tamer to handle).

So:

* Has anybody attacked this problem with modern technology? 
* Do you think that the above is a valuable question?
* I believe it is, because
    1. Every category that breaks Isbell condition (let's call it a "non-Isbell category" for the sake of brevity) seems quite nasty. And yet its homotopy theory can be well-understood. Isbell condition itself is stated in terms of the set theory of $\cal A$ and it is (unsurprisingly) linked to $\cal A$ having "nice" factorization systems ("nice" here means [proper](https://ncatlab.org/nlab/show/orthogonal+factorization+system)+something; did somebody explicitly prove this, maybe even Freyd?). So one can "foresee" if $\cal M$ will have a non-concrete localization proving that there is no homotopy-nice factorization system on $\cal M$ (a factorization system on $\cal M$ is homotopy-nice if it is an [homotopy FS](http://web.math.rochester.edu/people/faculty/doug/otherpapers/Bousfield_Fact.pdf) in the sense of Bousfield, and the FS induced by it on $\textsf{Ho}(\cal M)$ is nice).
    2. All these categories seem to exist (but I will be happy to see you disproving me, especially in the nontrivial cases):
         * a concrete category whose localization is concrete
         * a non-concrete category whose localization is concrete
         * a non-concrete category whose localization is non-concrete
         * a concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is non-concrete
         * a non-concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is concrete
         * a non-concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is non-concrete
         * a concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is non-concrete
    3. There's an interesting problem: every category is a quotient of a concrete category ([link](http://www.sciencedirect.com/science/article/pii/0022404971900041)). Is every category a *localization* of a concrete one?
* If $\textsf{Ho}(\cal M)$ is not concrete, there should be a property $P$ of $\cal M$ preventing $\textsf{Ho}(\cal M)$ to be Isbell. It seems obvious that every category $\cal N$ which is Quillen equivalent to $\cal M$ has $P$, and yet Freyd's technology seems to be rather context-specific, to the point that it's hard to believe that $P$ can be transported along the adjunction of a Quillen equivalence. Or maybe it is, under suitable assumptions?