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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?

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    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

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The problem with such a functor is group theoretical, not categorical. The problem arises because morphisms between groups need not map centers to centers. It doesn't have anything to do with universes, smallness, or foundational issues.

Consider for example $G=C_2$, $H=S_3$, $K=C_2$, and the maps $f\colon G\to H$ sending the nontrivial element of $G$ to $(1,2)$, and $g\colon H\to K$ by viewing $S_3/A_3$ as the cyclic group of order $2$.

Since $Z(G) = Z(K) = C_2$, and $Z(H) = \{1\}$, such a putative functor $\mathcal{F}$ would give that $\mathcal{F}(f)\colon C_2\to\{1\}$ is the zero map $\mathbf{z}$, and $\mathcal{F}(g)\colon \{1\}\to C_2$ is the inclusion of the trivial group into $C_2$. But $g\circ f=\mathrm{id}_{C_2}$, so $\mathrm{id}_{C_2} = \mathcal{F}(\mathrm{id}_{C_2}) = \mathcal{F}(gf) = \mathcal{F}(g)\mathcal{F}(f) = \mathbf{z}$ where $\mathbf{z}\colon C_2\to C_2$ is the zero map.

Thus, no such functor $\mathcal{F}$ can exist.

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    @Corey545: Why do you ask if Grp contains all groups? This example only uses $1$, $C_2$, and $S_3$ and surely you wouldn't call something the category of groups if it didn't at the very very least have a subcategory isomorphic to the category formed by those 3 groups and all homomorphisms between them.2012-06-19