# Every category is a localization of a concrete one?

$\require{AMScd}$Kučera, JPAA 1971 shows the remarkable result that every category is a quotient of a concrete one:

Given a category $\cal K$ there is a category $\check{\mathcal{K}}$ which is concrete and endowed with a (super-highly non unique) congruence $\sim$ such that the category having the same objects of $\check{\mathcal{K}}$ and $\hom_{\check{\mathcal{K}}}(K_1, K_2)/\!\!\sim$ as sets of morphisms is equivalent (in fact isomorphic) to $\cal K$.

Given that a particular example of a congruence is the homotopy relation on the cofibrant-fibrant objects of a model category, it seems natural to wonder if there is a finer result that classifies which categories are localization of concrete ones.

If I'm not wrong it is fairly easy to show that the class $\mathcal{W}_\sim$ of arrows in $\check{\mathcal{K}}$ defined by $$\Big\{ f\mid \exists g,g' : (fg\sim 1) \land (g'f\sim 1) \Big\}$$ is a 2-out-of-6 class. Then it seems natural to consider the localization $\check{\mathcal{K}}[\mathcal{W}_\sim^{-1}]$ and to consider the span $$\begin{CD} \check{\mathcal{K}} @>>> \check{\mathcal{K}}[\mathcal{W}_\sim^{-1}] \\ @VVV \\ \mathcal K \end{CD}$$ which seems to be canonically[1] built out of $\mathcal K$.

[1] I said that the construction given in Kučera's paper is not canonical; but I think you can refine it to let the correspondence $\mathcal{K}\mapsto \check{\mathcal{K}}$ a functor (and also that this functor is part of an adjunction).

• Perhaps Boris Chorny's answer (using work of Peter Freyd) at this link mathoverflow.net/questions/116702/… also gives a negative answer to your question. If the answer to your question was "yes" would there also be a sequence of reflections and coreflections contradicting Freyd's theorem? Apr 11 '17 at 17:23
• @DavidWhite No, it does not. As the category of homotopy types is precisely defined as a localisation of a concrete category (topological spaces, CW-complexes, the category of presheaves on a local test category, the category of small categories) at a class of its arrows. Apr 11 '17 at 19:38
• If the question is whether $\check{ \mathcal{K}}[\mathcal{W}^{-1}]$ coincides with $\mathcal{K}$, then the answer is "not necessarily". For example, suppose that $\check {\mathcal{K}}$ is the category $0^\to_\to 1$ and $\mathcal{K}$ is the quotient $0 \to 1$. Then $\mathcal{W}$ consists only of the identities and $\check {\mathcal{K}}[\mathcal{W}^{-1}] = \check {\mathcal{K}} \neq \mathcal{K}$. But perhaps Kučera's particular construction or a variation can be shown to be a localization. Apr 12 '17 at 22:15
• Another note: if $\mathcal{C}$ has a small generator, then there is no essentially surjective functor from $\mathcal{C}$ to the discrete category with as many objects as the size of the universe, never mind a quotient or localization functor. So the categories that Kucera constructs, though concrete, are likely still rather exotic in that they probably typically don't have generators. Maybe it would be interesting to ask in parallel what the quotients or localizations of categories with generators (or accessible categories, even locally presentable categories) can look like. Apr 13 '17 at 16:29