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As many people here will be aware, there is a debate within the foundations of category theory as to how to approach the discipline. One can either:

  1. Axiomatise the category of all categories directly using one's favourite axiomatisation.

  2. Interpret the categories set theoretically within an inaccessible cardinal. In this way we can mimic reasoning about the proper-class like entities of category theory with sets in the inaccessible model.

(A third method that deserves mention is to introduce a class theory (say $VNBG$ or $MK$) and do the interpretation there, but I'm not too concerned with this)

An obvious shortcoming of method (2) is that it is hard to talk about functor categories between large categories, as even one functor is already not an object in the inaccessible domain, so sets of these functors do not exist in the domain.

Method (2.) was then extended by Grothendieck to deal with this problem by proposing the following axiom:

[Grothendieck's Axiom] Every set $x$ is a member of some universe (In particular; for any universe $V$ there is a larger universe $V'$ such that $V \in V'$).

We can then interpret, say, functor categories between large categories of a given set theoretic model $V$ as sets within an extended model $V'$.

However, Grothendieck's method seems to have a serious shortcoming; it cannot account for the category of all categories (henceforth denoted by Cat) . For such a category does not occur as an object in any of these universes; there will always be objects and categories outside of any particular universe.

I am interested in exactly how much is at stake here. My question then is twofold:

  1. Is there a way of using a Grothendieck paraphrase to dodge this problem and nonetheless at least mimic talk of a category of all categories?

  2. Are there places in the literature where category theorists and other mathematicians make serious use of a category of all categories as an object of the theory (e.g. by showing that there is a functor from Cat to a different category)? Particularly welcome would be examples from applications of category theory to non-category theoretic mathematics (such as in the proof of Fermat's Last Theorem or in Grothendieck's work).

Many Thanks!

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    $\begingroup$ If we are content to work with the category of small sets instead of the category of all sets, why should we not be content to work with the category of small categories instead of the category of all categories? $\endgroup$ – Zhen Lin Aug 28 '13 at 17:54
  • $\begingroup$ I think I agree. If one is happy to draw the line at arbitrary points then there will be no problem in accounting for some portion of category theory. However I think it's a substantial philosophical problem for the person who wants to interpret the category of set as the sets under the first inaccessible that they don't respect the fact that category theoretic apply to all sets. My question is a technical one about whether or not it is possible to mimic a result about Cat using Grothendieck universes. So can talking about the cat of small categories mimic the talk about all categories? $\endgroup$ – Neil Barton Aug 28 '13 at 18:29
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    $\begingroup$ It depends on what you want to do. For some purposes one works with categories as certain algebraic structures much as one works with groups or rings or modules, and in those situations it is perfectly good to restrict to small categories. But in this day and age where Russell's paradox is so well known I very much doubt that there has been much serious work using the category of all categories. I also claim that any objection about the adequacy of the category of small categories will apply equally to the category of small sets. $\endgroup$ – Zhen Lin Aug 28 '13 at 18:46
  • $\begingroup$ I wholeheartedly agree that any objection about the adequacy of the category of small categories will apply equally well to the category of small sets. Do you think it would be fair to say then that the various attempts to axiomatise such a category (such as Lawvere's attempt) are only of interest to category theorists? Or is the axiomatisation useful for studying the category of small categories and other areas of mathematics too. Thanks very much for the patient explanations! $\endgroup$ – Neil Barton Aug 30 '13 at 19:28
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    $\begingroup$ The work of Lawvere and others in axiomatising the 2-category of small and "moderate" categories is mainly of interest to category theorists as a way of extracting a "formal category theory", much as category theory is a way of extracting "formal mathematics". $\endgroup$ – Zhen Lin Aug 30 '13 at 19:30

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