There are some basic discussions on the motivations of large categories and small categories [here] (On the large cardinals foundations of categories) and [here] (Large cardinal axioms and Grothendieck universes) and [here] (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 an discussion [here] (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 !

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    $\begingroup$ 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. $\endgroup$ – Zhen Lin Feb 19 '14 at 22:13
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    $\begingroup$ 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? $\endgroup$ – Fernando Muro Feb 19 '14 at 22:27
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    $\begingroup$ @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. $\endgroup$ – Zhen Lin Feb 20 '14 at 9:32
  • $\begingroup$ @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! $\endgroup$ – Fernando Muro Feb 20 '14 at 10:13
  • $\begingroup$ 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. $\endgroup$ – Zhen Lin Feb 20 '14 at 10:27

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.

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  • $\begingroup$ 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 ? $\endgroup$ – Tom Feb 21 '14 at 19:45
  • $\begingroup$ @Tom Thanks to google (!), i got this result : a small category is accessible if and only if it is idempotent complete (ncatlab.org/nlab/show/accessible+category). $\endgroup$ – Philippe Gaucher Feb 24 '14 at 14:28
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    $\begingroup$ 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? $\endgroup$ – Joel David Hamkins Mar 23 '14 at 15:12
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    $\begingroup$ 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.) $\endgroup$ – arsmath Mar 23 '14 at 15:41
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    $\begingroup$ 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. $\endgroup$ – Joel David Hamkins Mar 23 '14 at 16:08

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