5
$\begingroup$

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).

  • 7
    To 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
  • 0
    Pete: Thank you for pointing out my error. I mean small as in the class of objects being a set.2010-08-11
  • 0
    No 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
  • 0
    One 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
  • 1
    @Ringo: Are you a member of the Beatles?2010-08-11
  • 1
    Bryan, you mean essentially small, no?2010-08-13
  • 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
  • 0
    Questions under big-list are usually set to be community wiki. I think it should be community wiki.2010-11-20

5 Answers 5