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There are some basic discussions on the motivations of large categories and small categories: On the large cardinals foundations of categories, Large cardinal axioms and Grothendieck universes, Small model categories?.

As Mac Lane noted on large categories in categories for working mathematicians:

  1. One universe assumption is sufficient for all small sets and all small groups but does not provide category for all sets and all groups.
  2. Grothendieck gave a stronger assumption that for each universe a category of all those groups which are members of the universe. However it does not provide any category of all groups.
  3. Some proposals defined a category with a set-free terms such as axioms as first order axioms for the category of all sets. This includes elementary topos or logic tools.
  4. There are some ideas in the above discussions.
  5. For higher category theory, there is a discussion: Are grothendieck universes enough for the foundations of category theory?.

The two devices of the universes do not address the issue completely. Is the issue still open? References will be very appreciated!

Dmitri Pavlov
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Tom
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    The question is vague. What is "the issue"? The universe axiom is sufficient for most purposes – I have yet to encounter a situation where it is truly and unavoidably necessary to ask for a category of all sets or whatever. – Zhen Lin Feb 19 '14 at 22:13
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    Well, the category of all sets (or else) is sometimes necessary. I do have found such things in well established mathematical theories. Nevertheless, what's the problem of working with proper classes? – Fernando Muro Feb 19 '14 at 22:27
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    @FernandoMuro Working with classes is subtle. For instance, it is not possible to quantify over classes, let alone form collections of classes. In particular, there is no such thing as the category of all functors $\mathbf{Set} \to \mathbf{Set}$ if $\mathbf{Set}$ is genuinely the category of all sets. For the working mathematician, it is better to use the universe axiom than to worry about the finer details of logic. – Zhen Lin Feb 20 '14 at 09:32
  • @ZhenLin I know, but sometimes you want to say things about a proper class, and not about any set related to it, even working mathematicians do! – Fernando Muro Feb 20 '14 at 10:13
  • When using the universe axiom, all such statements $\phi$ are rephrased as, "all universes $\mathbf{U}$ satisfy the formula $\phi$ relativised to $\mathbf{U}$." And one can just treat $\mathbf{U}$-classes as sets in a larger universe. I have never seen a need to consider genuine proper classes in this context. – Zhen Lin Feb 20 '14 at 10:27
  • Nevertheless I have. That's what I meant. – Fernando Muro Feb 20 '14 at 11:02
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    Dear @Fernando, may I ask you to provide an example illustrating your claim? – Fred Rohrer Feb 21 '14 at 19:25
  • In the field of triangulated categories: well generated categories, Brown and Adams representability, etc. – Fernando Muro Feb 22 '14 at 00:29
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    Dear @Fernando, can you elaborate a bit? Far from being an expert but having just looked at a paper of yours concerning these things I am not yet convinced. (And maybe someone with more expertise (@Zhen?) could chime in?) – Fred Rohrer Feb 22 '14 at 06:01
  • @FredRohrer It's kind of complicated, I don't think I'd be able to convince Zhen Lin (nor you, if you're also skeptic). It's not my intention, either. I always though I could live a mathematical life without set theoretical problems, and one day I found I couldn't. I'd recommend Neeman's book on triangulated categories. Or any application of Vopenka's principle to homotopy theory. – Fernando Muro Feb 22 '14 at 09:36
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    @FernandoMuro Well, if you want to use Vopěnka's principle, that can be made to fit in the universe-ful setup as well: just posit that each set is a member of some universe that satisfies the relativised version of Vopěnka's principle. But I prefer not to assume large cardinal axioms to make set-theoretical difficulties of that nature go away; the purpose of the universe axiom to repair a deficiency in first-order logic. – Zhen Lin Mar 23 '14 at 13:25
  • @ZhenLin: I don't understand your statements about first-order logic. You can quantify over classes, in Goedel-Bernays or Kelley-Morse set theory. If you really want to, you can write a first-order formulation that allows collections of classes (this is the formulation that "Abstract and Concrete Categories" has in mind, though the book doesn't write it out explicitly). – arsmath Mar 23 '14 at 15:32
  • Anyway, the axiom of universes is a large cardinal axiom, just a small large cardinal axiom. – arsmath Mar 23 '14 at 15:33
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    @arsmath Yes, I am perfectly aware. I did write a whole article on the subject, after all. In my mind, the purpose of the universe axiom is to allow us to treat sets, classes, collections of classes, collections of collections of classes, etc. all on the same basis. – Zhen Lin Mar 23 '14 at 16:07

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This should be a comment, not an answer but it is too long ! This issue is not open. On the contrary, it is well-known. Unlike what is commonly believed, ZFC alone is not sufficient to develop category theory. With ZFC and three Grothendieck universes $\mathcal{U}_1\in \mathcal{U}_2\in \mathcal{U}_3$, category theory can be built. So ZFC and three strongly inaccessible cardinals. See for example M. Makkai, R. Paré, Accessible Categories : The Foundations of Categorical Model Theory, Contemporary Mathematics 104, in the introduction. For some parts of category theory, the distinction between set and class is sufficient. Bernays-Gödel set theory is used in the book Locally Presentable Accessible Categories by Jiri Adamek and Jiri Rosicky.

Philippe Gaucher
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  • Model theory has well known algebraic techniques and logic tools to address the similar issue but for the possibility of accessible category, is any small category accessible ? – Tom Feb 21 '14 at 19:45
  • @Tom Thanks to google (!), i got this result : a small category is accessible if and only if it is idempotent complete (http://ncatlab.org/nlab/show/accessible+category). – Philippe Gaucher Feb 24 '14 at 14:28
  • That is not what I meant. – Tom Feb 25 '14 at 16:14
  • @Tom I think that nobody understands what you mean, just because there is no issue, even for computers. Indeed, the proof assistant MIZAR uses a variant of set theory called Tarski–Grothendieck set theory. Roughly speaking in this extension, there are as many Grothendieck universes as necessary. – Philippe Gaucher Feb 26 '14 at 16:55
  • This is a strong and quick comment but the argument is involved with more issues. Did you pay attention to my earlier comment on model theory ? – Tom Feb 27 '14 at 15:07
  • Do you mean the characterization of accessible categories by the categories of models of a basic theory ? You want to consider the category of all sets or of all groups, whatever the universes they belong to. That is not possible in ZFC: the set of all sets leads to a contradiction. And that is not necessary for mathematicians. So where is the issue ? – Philippe Gaucher Feb 28 '14 at 09:02
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    Philippe, do you make the assertion, "ZFC alone is not sufficient to develop category theory" as a mathematically precise claim, for example in terms of consistency strength or an independence result? If so, could you say more precisely what you mean? Or perhaps you merely make the assertion as an informal observation? – Joel David Hamkins Mar 23 '14 at 15:12
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    I'm skeptical that there is any agreement on this ZFC + 3 universes business. As far as I can see, most theorems in a standard introductory category theory book can be formulated in ZFC alone. (There are a few results whose proofs require the existence of the skeleton of an arbitrary category, which doesn't hold in ZFC, but holds in a conservative extension.) – arsmath Mar 23 '14 at 15:41
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    If you think that 3 universes are required, then the way to prove this is to show that there is some desired element of the theory that has the same consistency strength as ZFC+3 inaccessible cardinals. – Joel David Hamkins Mar 23 '14 at 16:08
  • @JoelDavidHamkins My statement "ZFC alone is not sufficient to develop category theory" was an informal observation. It lies on the fact that one has to deal sometimes with "big" objects and "superbig" objects in the proofs. Maybe there is a way to avoid this big and superbig objects. The book "Homotopy Limit Functors on Model Categories and Homotopical Categories" from Dwyer, Hirscchhorn, Kan and Smith is another example of use of universes. – Philippe Gaucher Mar 25 '14 at 03:57
  • @JoelDavidHamkins If you think that it is just a convenient way and that there is a way to stay inside ZFC, I would be curious to know more about this subject. My informal observation comes also from the fact that an assistant proof like MIZAR uses Grothendieck universes. So that made me think that people working on reverse mathematics had proved that Grothendieck universes were necessary for the formal development of some parts of mathematics. It is all I know about this subject. – Philippe Gaucher Mar 25 '14 at 03:59
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    @PhilippeGaucher It seems that few or even none of the applications of Grothendieck universes actually require them, and one can easily get by with a variety of weaker universe concepts that are provable in ZFC or much less. See my answer here for a summary of what I mean: http://mathoverflow.net/a/28913/1946. These more nuanced universe concepts take a little more care to work with, but since they are weaker in consistency strength, they show that the inaccessible cardinals are not required for the applications. – Joel David Hamkins Mar 27 '14 at 01:18