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First of all, I have to mention that I'm truly sorry if this question would seem inappropriate for this site for some people. Still, I think it is better to ask here rather on math.stackexchange.

I recently grasped basic notions of Grothendieck universes for category theory. We most use the axioms of Zermelo and Fraenkel, Axiom of Choice and an additional axiom saying that any set is contained within some universe. What is more, I understood that the purpose of universes, intuitively, is that we can "do set theory" within the universe. That is, we can construct new sets out of old ones using the axioms of Zermelo and Fraenkel, and still stay within universe. But my understanding may be false, please, correct me, if I'm wrong. Many of the sources I consulted are either very technical or very brief, so my understanding of the matter is somewhat vague.

Still, as of now, I don't think I understand what Grothendieck universes and the axiom (the one about every set being in a universe) brings to the table of category theory. Classically, people wanted to talk, for example, about all categories of sets with (or without) structure (e.g. $\mathrm{Grp}, \ \mathrm{Ring}, \ \mathrm{R-Mod}$ or even $\mathrm{Set}$). So, if we use classes, the collections of objects in the aformentioned categories would be proper classes. Now, if we use universes, we fix a Grothendieck universe $\mathbb{U}$ and assume that collections of objects and morphisms in any category are actually sets. Then we call the members of $\mathbb{U} \ \ \mathbb{U}$-sets, the subsets of $\mathbb{U} \ \ \mathbb{U}$-classes, and we call a $\mathbb{U}$-class proper if it is not a member of of $\mathbb{U}$.

After that, we call a category $\mathrm{C}$ a $\mathbb{U}$-small if the set of objects and the set of morphisms of $\mathrm{C}$ are $\mathbb{U}$-sets. And we call $\mathrm{C}$ a locally small if the set of objects and the set of morphisms of $\mathrm{C}$ are $\mathbb{U}$-classes, and $\forall X,Y \in \mathrm{C}: \ \mathrm{Hom_C} (X,Y)$ is a $\mathbb{U}$-set.

But I have to wonder what is the motivation? I realise that in the end it is supposed to ease the foundations of category theory on set theory by erasing the need on distinguishing between sets, classes, conglomerates and whatever the "higher" notions are called (sorry, I haven't heard of them). But how is this supposed to make up for the fact that we don't study "the whole" category (such as the category of all sets, for example) anymore? And what is the motivation behind the notion of Grothendieck universe in respect of category theory? So, we can "do set theory" in a universe, and what it does for us as far as category theory goes?

To sum up: my question is what exactly the use of Grothendieck universes gives us so we don't need to study categories with proper classes of objects and morphisms, or proper conglomerates of objects and morphisms, or other collections of objects and morphisms that are not sets? If we assume that any category has only a set of objects and morphisms, but this time use the help of Grothendieck universes, why this works? Why can we forsake collections of elements that are too big to be a set in favor of sets and only sets if we fix a universe $\mathbb{U}$ and study whether sets of objects and morphisms of a category belong to $\mathbb{U}$? What is the "magic" of a Grothendieck universe?

P.S. I consulted several sources before asking this question, they were helpful to me in other aspects, but in the end they didn't resolve my confusing on this specific matter. In particular, I consulted Borceux'. Mac Lane's and Kashiwara's (coathored with P.Schapira) books on category theory, the articles by D.Murfet and M.Shulman. I know there is supposed to be an extensive exposition in SGA 4, but, unfortunately, I can't read French.

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    $\begingroup$ An example where it is useful to either have universes or use some appropriate set theory is to prove a functor is fully faithful and essentially surjective if and only if it is an equivalence. One uses choice on the object set to construct the quasi-inverse and so it is convenient to have them form a set. $\endgroup$ – Benjamin Steinberg Nov 23 '16 at 16:58
  • $\begingroup$ I don't quite understand what you're asking, but I think you might find some motivation and enlightment in Mike Shulman's post on universe polymorphism and typical ambiguity. $\endgroup$ – Andrej Bauer Nov 23 '16 at 17:07
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    $\begingroup$ You should look at SGA 4 (especially Bourbaki's Appendix) even if you can't read French. $\endgroup$ – Fred Rohrer Nov 23 '16 at 20:48
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    $\begingroup$ See my post mathoverflow.net/a/28913/1946 on how uses of Grothendieck universes can often be softened without loss to various weaker universe concepts. $\endgroup$ – Joel David Hamkins Nov 24 '16 at 0:23
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    $\begingroup$ Dear Jxt921, I just remembered some old notes of mine that are more or less a translation of the basic universe stuff in SGA. $\endgroup$ – Fred Rohrer Nov 24 '16 at 15:44
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Universes provide a convenient framework for stacking many levels of largeness. But a class in NBG cannot contain any other class. The most basic application which shows that universes are more convenient is that the category of functors $[\mathcal{C},\mathcal{D}]$ between two $U$-categories $\mathcal{C},\mathcal{D}$ is again a $V$-category if $U \in V$ are two nested universes. You cannot even write down this construction in NBG (unless you use inaccessible cardinals, which then is more or less equivalent to the universe approach).

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    $\begingroup$ I'm beginning to develop a vague understanding and intuition. Still, why can we forsake such classical categories as $\mathrm{Set}, \ \mathrm{Grp}, \ \mathrm{Ring}, \ \mathrm{Top}$ in favor of the category of all $\mathbb{U}$-sets, the category of all $\mathbb{U}$-groups, the category of all $\mathbb{U}$-rings and the category of all $\mathbb{U}$-topological spaces? Don't we need the "entire" category which would have proper classes of objects and morphisms in NBG? $\endgroup$ – Jxt921 Nov 24 '16 at 8:00
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    $\begingroup$ You won't be able to do many constructions with this "entire" category. And no, we don't really have to look at them. Also, it is common practice to just write "groups" when one uses U-groups if the context permits this. So nothing really changes except you make "large" constructions such as functor categories, in particular sheaf categories, etc. where one usually adds a short remark about universes. $\endgroup$ – HeinrichD Nov 24 '16 at 8:13

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