Sometimes one encounter the requirement that the objects of a category needs to be a set. What if it was not, could you provide examples of what could go wrong? (One example per answer).
What's so special with small categories?
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big-list
set-theory
category-theory
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7To be clear, a category is **small** if its class of *objects* is a set. A category is sometimes called **locally small** if for every pair of objects $A,B$, the *morphisms* from $A$ to $B$ form a set. (However, the most standard definition of a category requires this anyway.) Do you mean to ask about small, or locally small, categories? – 2010-08-11
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0Pete: Thank you for pointing out my error. I mean small as in the class of objects being a set. – 2010-08-11
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0No problem, Ringo. As I understand it, the disadvantages of non-small categories primarily lie in *higher* category theory, where they can lead to set-theoretic difficulties. But I'm sure someone can do better than that... – 2010-08-11
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0One more little point is that most of the things I read don't require small, they require svelte, which means they have a small skeleton. This point is nearly irrelevant but has some substance. Another common axiom that is assumed is Grothendeick Universes. – 2010-08-11
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1@Ringo: Are you a member of the Beatles? – 2010-08-11
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1Bryan, you mean essentially small, no? – 2010-08-13
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0@Pete: as I see it, I wouldn’t say that higher category theory itself has more problems with size issues. It’s more, I think, that the places one needs to confront size issues are often the same places one meets higher-categorical issues. – 2010-11-20
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0Questions under big-list are usually set to be community wiki. I think it should be community wiki. – 2010-11-20