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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?

Mozibur Ullah
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    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. – Zhen Lin Dec 01 '12 at 19:32
  • Related: http://mathoverflow.net/questions/24552/what-interesting-nontrivial-results-in-algebraic-geometry-require-the-existence-o/28913#28913 – Joel David Hamkins Dec 01 '12 at 19:43
  • @Lin: So you're saying that grothendieck universes are simply a useful convenience, but not neccessary for foundations? – Mozibur Ullah Dec 01 '12 at 19:59
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    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) – Feldmann Denis Dec 01 '12 at 20:50
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    @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. – Zhen Lin Dec 01 '12 at 21:43
  • @Lin: My concern is that category theory is being mooted as a foundation for mathematics, in the way that ZFC now does. That is you can build ZFC via category theory. But if ZFC is being extended in some ways, ie through large cardinals. How does foundational category theory model that? – Mozibur Ullah Dec 01 '12 at 23:30
  • @Denis: thanks for the clarification. – Mozibur Ullah Dec 01 '12 at 23:34
  • I don't think it's really accurate to regard category theory as a "foundation" for mathematics in the same sense that ZFC is. There are alternatives to ZFC, some of which are motivated by category theory, but category theory itself doesn't really play the same role. – Mike Shulman Dec 23 '12 at 03:03
  • @Shulman: I wouldn't expect a different foundation to play exactly the same role as the original, but one would expect there to be a great deal of commonality. Does this mean that ETCS is entirely standalone? So although its inspired by category theory that scaffolding can be taken away. I find this point a little confusing: why do this? Isn't there an 'elementary theory of categories' that doesn't rely on ZFC (I thought Maclane tries to do this in his book) then why not keep to the natural order of 'inspiration'? Unless of course there is no such elementary theory as it runs into difficulties – Mozibur Ullah Jan 22 '13 at 04:22

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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.

Daniel Schäppi
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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.

Fred Rohrer
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