16
$\begingroup$

The category $\mathbf{Set}$ contains as its objects all small sets and arrows all functions between them. A set is "small" if it belongs to a larger set $U$, the universe.

Let $\mathbf{Grp}$ be the category of small groups and morphisms between them, and $\mathbf{Abs}$ be the category of small abelian groups and its morphisms.

I don't see what it means to say there is no functor $f: \mathbf{Grp} \to \mathbf{Abs}$ that sends each group to its center, when $U$ isn't even specified. Can anybody explain?

  • 4
    The claim you're asking about should be true for every universe $U$ and should have nothing to do with universes; there should be a counterexample using a few finite groups.2012-06-14
  • 8
    General advice: don't worry about universes. They're only there to pacify logicians and set theorists..2012-06-14
  • 11
    Well, there is a time and a place to worry about universes, but this isn't it.2012-06-14
  • 0
    This is not a category problem, you should look for why this does not make sense by group homomorphisms.2012-06-15
  • 1
    There also is this nice post http://qchu.wordpress.com/2012/02/06/centers-2-categories-and-the-eckmann-hilton-argument/ by Qiaochu Yuan above2012-06-15
  • 0
    If we restrict our category to groups with monomorphisms, and $f:G\rightarrow H$ is a homomorphism of groups, then it does seem to be the case that $f^{-1}(Z(H)) \subseteq Z(G)$. Viewing this containment as a monomorphism, this does give a contravariant functor from Groups with monos to Ab with monos.2015-01-01

1 Answers 1