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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.

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    $\begingroup$ you can look at : arxiv.org/abs/1704.00303 but as far as I know, not much more is known $\endgroup$
    – Simon Henry
    Commented Dec 25, 2018 at 11:41
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    $\begingroup$ @JohnBaez: Ivan recently gave a talk at MPIM on this subject: math.muni.cz/~diliberti/Talk/Golem.pdf and has a section in it on his research plan (§1 of math.muni.cz/~diliberti/memoir.pdf ) $\endgroup$
    – fosco
    Commented Dec 25, 2018 at 12:11
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    $\begingroup$ It seems likely to me that there is a functors from $(\infty,n)$-categories to spaces faithful on the homotopy categories. For example sending a category to the disjoint union of its spaces of $k$-cells for $k \leqslant n$, $\endgroup$
    – Simon Henry
    Commented Dec 25, 2018 at 13:22
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    $\begingroup$ To clarify: the question only makes sense if "category" means "locally small category". Kucera showed that every category is a localization of a concrete category. Velebilova gave conditions under which a localization of a concrete category is concrete. Perhaps thinking about her conditions could give a stratification of categories by concreteness. Freyd also gave necessary and sufficient conditions for concreteness in "On the concreteness of certain categories". $\endgroup$
    – Tim Campion
    Commented Dec 25, 2018 at 20:35
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    $\begingroup$ There is also work by Pultr and Trnkova on the preorder on categories given by full embeddability. Keyword "universal categories". $\endgroup$
    – Tim Campion
    Commented Dec 25, 2018 at 21:21

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