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Grothendieck universes are equivalent to ZFC+a strongly inaccessible cardinal. This is low on the large cardinal axiom list. Is it enough to place category theory on a firm foundational basis, and how about higher category theory, does it remain enough?

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    $\begingroup$ Category theory does not need any large cardinals to be put on a firm foundational basis – and if you are willing to give up certain transfinite constructions, you won't even need the full strength of ZFC. The use of inaccessible cardinals is essentially just a convenience. $\endgroup$ – Zhen Lin Dec 1 '12 at 19:32
  • $\begingroup$ Related: mathoverflow.net/questions/24552/… $\endgroup$ – Joel David Hamkins Dec 1 '12 at 19:43
  • $\begingroup$ @Lin: So you're saying that grothendieck universes are simply a useful convenience, but not neccessary for foundations? $\endgroup$ – Mozibur Ullah Dec 1 '12 at 19:59
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    $\begingroup$ Actually, the Grothendieck axiom of universe U is much stronger than existence of a strongly inaccessible cardinal, being equivalent to the exoistence of a proper class of inaccessibles (but it is weaker than the existence of a 1-inaccessible cardinal) $\endgroup$ – Feldmann Denis Dec 1 '12 at 20:50
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    $\begingroup$ @Mozibur: Category theory is fundamentally the study of a certain kind of essentially algebraic structure. It is quite possible to do non-trivial things with categories that only have a set of objects and a set of morphisms – but people often want to apply category theory to study things like "all sets" or "all groups", and this is where it becomes convenient to posit that our set-theoretic universe can be embedded in another. Even then, if one is not too greedy, many common constructions can be formalised as theorems or meta-theorems in NBG or ZFC. $\endgroup$ – Zhen Lin Dec 1 '12 at 21:43
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Mike Shulman wrote a nice expository paper on set theoretical foundations for category theory

http://arxiv.org/abs/0810.1279

In Section 6 he explains the difficulties of working with large categories using just ZFC, and he discusses various ways to deal with these size issues. Some of these do not assume the existence of an inaccessible cardinal.

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I suggest you have a look at Grothendieck's SGA 4, Exposé I, where a lot of category theory is developed (and applied) on the basis of Bourbaki set theory plus Grothendieck's axioms UA and UB about universes.

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