For a small abelian category in which every object is also a set, consider its localization with respect to a Serre subcategory (thus a quotient category), is it true that under this localization functor $T$, the image of an object is still a set? I wonder if it is always possible to turn this functor $T$ into an endofunctor $F$ such that its image is equivalent to the localized category, as it is considered in triangulated categories where a localization functor is defined as an endofunctor together with a natural transformation satisfying certain properties. In this sense, the first question seems to be resolved naturally.

2$\begingroup$ It sounds to me like the assumption that "every object is also a set" is misguided. What would be an example of a small abelian category in which not every object is a set? The very notion of "small" presupposes set theory, and so the objects of a small category are automatically sets. If you are considering a set theory with urelements then we can get rid of them by moving to an isomorphic category. $\endgroup$ – Andrej Bauer Oct 14 '15 at 20:22

$\begingroup$ Is it possible to have an (essentially) small category in which each object is not a set? $\endgroup$ – user81489 Oct 14 '15 at 21:05

2$\begingroup$ Let us suppose we are working in ZermeloFraenkel set theory without urelements. Then everything is a set. I do not even understand why you worry about the settheoretic nature of the objects. Why do you even care whether the objects are sets, bananas or coconuts? $\endgroup$ – Andrej Bauer Oct 14 '15 at 21:33

$\begingroup$ My problem focuses on the localized category, and I wonder when applied the localization functor $T$ the image $T(M)$ of an object $M$ (which is a set) remains a set or not. In the category of finitely generated abelian groups, the localization with respect to the Serre subcategory generated by $\mathbb{Z}/(2)$ say corresponds to the functor given by tensoring with $\mathbb{Z}_{(2)}$. In particular, the image of every abelian group is still a set, the elements are given by equivalence classes of fractions. So I wonder if it is true in general that $T(M)$ is a set for arbitrary localization. $\endgroup$ – user81489 Oct 14 '15 at 21:46

2$\begingroup$ Do you mean rather that every object of the category has an underlying set? $\endgroup$ – David Roberts Oct 15 '15 at 1:40
Perhaps @user81489 means the following. Suppose given a concretizable category (note: every small category $\mathcal C$ is concretizable, by the functor $\prod_{C\in \mathcal C} \hom(C,)$; see also this post on SBS.). Is its localization with respect to a Serre subcategory also concretizable?
The localization of small category with respect to some subcategory is necessarily small, so in that case the answer is "yes". On the other hand, I believe that one can modify Freyd's theorem that the homotopy category of topological spaces is not concretizable to the abelian setting to produce a counterexample when smallness is dropped.

$\begingroup$ Yours is a better phrased question. Also thanks for providing the post. As you mentioned, why is the localized subcategory necessarily small? Is it simply because the way of a morphism chosen is depended on a set of objects? $\endgroup$ – user81489 Oct 15 '15 at 3:03

1$\begingroup$ Yes. A morphism in $\mathcal C[\mathcal W^{1}]$ is an equivalence class of zigzag paths in $\mathcal C$ whose "zag" components are in $\mathcal W$. If $\mathcal C$ is small, there are only a set of such zigzags, and hence only a set of equivalence classes. $\endgroup$ – Theo JohnsonFreyd Oct 15 '15 at 17:04