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In some books on category theory (for example, in J.Adámek, H.Herrlich, E.Strecker "Abstract and concrete categories...") the authors use the idea of "big sets" ("conglomerates" or "collections") which can contain classes (as far as I understand, in the Goedel-Bernays sense) as elements, and they formulate the "generalized axiom of choice", where it is stated that the choice function exists (not only for families or classes of sets, but also) for families of classes (indexed by elements of those "big sets"). This approach allows to prove, in particular, the existence of a skeleton in each category, and some other useful things.

This generalization of the axiom of choice is also mentioned In Wikipedia: http://en.wikipedia.org/wiki/Axiom_of_global_choice (as the "strong form of the axiom of global choice").

I wonder if there are any texts with the justification of this trick? The references I found (in particular, those mentioned in Wikipedia) give justification only for usual axiom of choice (for families of sets or for classes of sets, but not for "conglomerates of classes"). So actually I can't understand whether, for example, the existence of a skeleton, is true for all categories (in some interpretation of set theory) or for some special ones... Similarly the other corollaries of this "global axiom of choice" look doubtful for me. I would be grateful if anybody could clarify this.

UPDATE 21.09.2012

From the comments I see that there is a risk of misunderstanding, so I want to explain that by justification I mean an accurate (rigorous) definition of the new tool together with the analysis of whether it is compatible with the other tools of the theory.

As an illustration, in the case of the usual axiom of choice (I mean its "weak form", in terms of Wikipedia), there are many textbooks (I can recommend E.Mendelson "Introduction to mathematical logic" or J.Kelly "General topology", the appendix), where the fundamental objects of the theory (in this case, the classes) are accurately introduced (here, axiomatically) and the necessary constructions (like functions) are rigorously defined in the theory. This makes possible to give rigorous formulation to the axiom of choice (again, to its "weak form") inside the theory, and moreover, this presentation of a new axiom is followed by a thorough investigation of whether it contradicts to the previous axioms of the theory. Only after receiving the answer that no contradictions can appear (in fact, a more strong thing is true: the new axiom is independent from the others, that was the result by P.Cohen) mathematicians can use this new axiom without worrying that something is wrong here.

So my question is whether there is something similar for the "strong form of the axiom of choice"? Is it possible that nothing lies behind these words? On the contrary, if there is a justification, where can I read about it?

UPDATE 21.04.2013

Dear colleagues, from what I learn on this subject in the textbooks which I found, in Wikipedia and here in MO, I deduce that what people call "axiom of global choice" is just the usual axiom of choice (as it is presented in Kelly's book) applied to some special classes of sets arising in consideration of what is called the Grothendieck Universe. It's a puzzle for me

1) why people call this special case "a stronger form of the axiom of choice", and

2) why they don't want to give references, where this construction is accurately introduced.

With the aim to accelerate the clarification of this question, I now nominated for deletion the article in Wikipedia devoted to his topic: http://en.wikipedia.org/wiki/Wikipedia:Articles_for_deletion/Axiom_of_global_choice. As I wrote there, I don't exclude that the partisans of the idea will rewrite the article in Wikipedia for endowing "global choice" with some mathematical sense, but you should agree that in its present form this article and the other mentionings of "global choice" available for external observers, look indecently vague. I invite all comers to share their opinion here or at the Wikipedia page.

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One way of putting it is this: the axiom of choice for sets is equivalent to the statement that every small category has a skeleton. Since the definition of category is first order, you could consider the axiom: Every model of the category axioms has a skeleton. This is then prior to any choice of ambient set theory. –  David Roberts Sep 20 '12 at 7:27
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What exactly do you mean by "justification" of an axiom? I'd use that phrase to mean the pre-axiomatic intuitive ideas that lead me to regard the statement as a reasonable axiom to include in my theory. For this purpose, it seems to me that whatever intuition leads you to accept the axiom of choice for sets would probably do the same for classes, conglomerates, and whatever higher entities you include in your theory. –  Andreas Blass Sep 20 '12 at 13:25
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I don't know that much about set theory, but I guess to get a workable theory of conglomerates (in ZF+something) we need some large cardinal axiom and then the axiom of choice is simply the usual axiom of choice, applied to very large sets then. And if the most intriguing part is "how can a class (which is not a set) be an element of someting else?", then this has little to do with the axiom of choice. –  Michael Greinecker Sep 21 '12 at 10:35
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Sergei: The main reason why you're not getting an answer is not necessarily the meaning of "justification" but the meaning of "conglomerate." None of the standard set theories (ZF, NBG, MK) admit those. Some more esoteric theories do (e.g. Ackermann) but since these can be wildly different from each other, you really need to say which one you're using before any kind of serious analysis can be done. If you don't know which one you are using then you can ask a separate question to figure that out first. –  François G. Dorais Sep 21 '12 at 12:21
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Indeed, I am also appalled that my civilization still can't answer vague questions about undefined objects. –  François G. Dorais Sep 22 '12 at 6:47
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3 Answers

Dear Sergei, you might be interested in first reading Bourbaki's Théorie des ensembles (at least chapters I--III) and then have a look at section 0 and the appendix of SGA 4.I. This gives a slightly different approach using Hilbert's almighty symbol $\tau$.

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Appendix of SGA? What is it? In which translation is this? English? –  Sergei Akbarov Apr 21 '13 at 10:35
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Dear Sergei, I refer to SGA 4, Expose I, Appendix ("Univers" by N. Bourbaki). I do not know of any translation of SGA, so you have to go with the french original. –  Fred Rohrer Apr 21 '13 at 10:42
    
Pedantic remark: Hilbert's symbol was actually $\varepsilon$. en.wikipedia.org/wiki/Epsilon_calculus –  François G. Dorais Apr 21 '13 at 18:31
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Here at least is the usual justification for moving from AC for sets to what is normally called the global axiom of choice, which asserts that there is a class well-ordering of the (first-order) universe.

Theorem.

  1. The global axiom of choice, when added to the ZFC or GB+AC axioms of set theory, leads to no new theorems about sets. That is, the first-order assertions about sets that are provable in GBC are precisely the same as the theorems of ZFC.

  2. Furthermore, every model of ZFC can be extended (by forcing) to a model of GBC, in which the global axiom of choice is true, while adding no new sets (only classes).

  3. In particular, the global axiom of choice is safe in the sense that it will not cause inconsistency, unless the underlying system without the global axiom of choice was already inconsistent.

Proof. Suppose that $M$ is any model of ZFC. Consider the class partial order $\mathbb{P}$ consisting of all well-orderings in $M$ of any set in $M$, ordered by end-extension. As a forcing notion, this partial order is $\kappa$-closed for every $\kappa$ in $M$, since the union of a chain of (end-extending) well-orderings is still a well-order. If $G\subset\mathbb{P}$ is $M$-generic for this partial order, then $G$ is, in effect, a well-ordering of all the sets in $M$. Furthermore, one can prove by the usual forcing technology that the structure $\langle M,{\in},G\rangle$ satisfies $\text{ZFC}(G)$, that is, where the predicate $G$ is allowed to appear in the replacement and other axiom schemes.

Essentially, what we've done is add a global well-ordering of the universe generically. And since the forcing was closed, no new sets were added, and so $M[G]$ has the same first-order part as $M$.

It follows now that GBC is conservative over ZFC for first-order assertions, since any first-order statement $\sigma$ that is true in all GBC models will be true in $M[G]$ and therefore also in $M$, and so $\sigma$ is true in all ZFC models as well. QED

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I am afraid, this is too technical for me. Are you saying that it is possible to extend the Gödel-Bernays theory in such a way that the axiom of choice becomes more powerful, it can be applied to classes? –  Sergei Akbarov Apr 13 '13 at 16:39
    
No, I am saying that the extension of ZFC to Gödel-Bernays GBC theory, which includes the global axiom of choice (which is the axiom of choice applied to classes), is known to be completely safe and does not create any new theorems that are purely about sets. (Of course, it does create new theorems about classes, since global AC is such a new theorem about classes, that is not provable in GB+AC without the global axiom of choice.) You can find further discussion of this in any graduate text that treats Gödel-Bernays set theory. –  Joel David Hamkins Apr 13 '13 at 17:12
    
Which text namely? –  Sergei Akbarov Apr 13 '13 at 17:17
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I might add that this is completely standard material when considering Goedel-Bernays set theory, and so if you do find this "completely unfamiliar", then it may be appropriate for you to adopt a less strident tone. –  Joel David Hamkins Apr 21 '13 at 11:30
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In particular, the Wikipedia page on global choice en.wikipedia.org/wiki/Axiom_of_global_choice seems to me currently to be completely fine, presenting the standard and familiar facts about it as they are usually understood, including the conservation result that I prove above. Please do not delete that Wikipedia page, and please remove your request for deletion. –  Joel David Hamkins Apr 21 '13 at 11:40
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I think that you should place yourself in ZFC+ existence of strongly innaccessible cardinals.

Then the existence of a strongly inaccessible cardinal provides you a universe as in Borceux's Handbook of Categorical algebra.

Then, what you call sets are elements of the universe, and what you call classes are the subsets of the universe, but they are still sets in the set theoretic sense, so you can apply choice.

EDIT: clarification

The problem of category theory is that we want to have the category Set of all sets to actually be a category.

Since there is no set which contains all sets, we can't ask a category to have a set of objects, or Set will no more be a category. That's why in the first place we define the collection of object to be potentially wider than a set: we ask it to be a class.

The point is that you can avoid the difficulty differently, by limiting yourself to a rich enough set of sets, which should contains "everthing that you can be interested in".

This is the concept of a Grothendieck universe, see http://ncatlab.org/nlab/show/Grothendieck+universe or Borceux's Handbook of Categorical algebra for a definition.

Existence of Grothendieck universes turns out to be equivalent to the existence of strongly inaccessible cardinals (here we are in ZFC), and this existence axiom has been studied in set theory (I'm not a specialist of that at all).

So you place yourself in ZFC + existence of strongly inaccessible cardinals, and you take a universe $U$. Call the elements of U the "U-sets" and the subsets of U the "U-classes".

Then, define a category $\mathscr C$ (in the universe $U$) to be a triple $(Obj~ \mathscr C, Mor~ \mathscr C, \circ)$ with $Obj~ \mathscr C \subseteq U$ and $\mathscr C(A,B) \in U$ for all $A,B \in Obj~ \mathscr C$, satisfying the usual axioms of a category.

So now, your U-classes are indeed sets of ZFC (as subsets of the set U), so you can use the axiom of choice in them, without bothering.

I am not sure if it is what you were looking for, but it is what I personally have in mind when I am using the axiom of choice to choose in a collection of objects, in category theory.

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Dimitri, I did not undertand you. Is this written anywhere? If not, can you contact me to explain what you write? (Or explain this here.) –  Sergei Akbarov Apr 8 '13 at 5:18
    
I hope my edit made the ideas I tried to express clearer. –  Dimitri Zaganidis Apr 13 '13 at 15:39
    
Dimitri, is it possible to translate this into the language of the Gödel-Bernays set theory? –  Sergei Akbarov Apr 13 '13 at 16:15
    
As far as I understand, $U$ must be a set here, is it? If yes, then I don'see any profits. If the "axiom of global choice" is just the usual axiom of choice applied to families of $U$-classes (which are families of sets), then why do people introduce the very term "axiom of global choice", and claim that "the axiom of global choice is a stronger variant of the axiom of choice which applies to proper classes as well as sets" (en.wikipedia.org/wiki/Axiom_of_global_choice). This is evidently not true in the situation you describe. –  Sergei Akbarov Apr 20 '13 at 8:05
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