Does the set Hom(X,Y), in any category, always form an Abelian group?

Question

Given any category C and any two C-objects X and Y, is it always the case that the set Hom(X, Y) of morphisms from X to Y form an Abelian group? If so, and to be clear, is this always the case regardless of whether or not C itself is Abelian, Ab?

Answer 1

No. In general, there's no operation on $\mathbf{Hom}(X,Y)$. To be specific: if we're working in the category of sets, what operation are you putting on $\mathbf{Hom}(\{\emptyset\},\{\emptyset,\{\emptyset\}\})$? Obviously, you can put an abelian group structure on any set you like (or rather, I'll say on any reasonable set, to avoid messing around with large cardinal stuff), and Hom-sets are always sets, so you could always put an abelian group structure on it, it's just not useful and doesn't tell you anything about the categories or the objects that you didn't already know from properties of the underlying set.

It is true for Abelian categories, however (you can embed your category in $\mathbf{Ab}$, and use the operations there).

Answer 2

You can put an abelian group structure on pretty much any set you like, but it won't necessarily give you anything meaningful.

A category $\mathcal{C}$ whose hom sets $\mathcal{C}(X,Y)$ are endowed with an abelian group structure $(+,0)$. that interacts nicely with the categorical structure of $\mathcal{C}$, is called an $\mathbf{Ab}$-enriched category (or one of a number of other names, such as a ringoid).

By 'interacts nicely', I mean that composition $\circ : \mathcal{C}(X,Y) \times \mathcal{C}(Y,Z) \to \mathcal{C}(X,Z)$ is bilinear with respect to the three group operations involved.

This is a fairly strong condition that is not enjoyed by most categories. For example, in an $\mathbf{Ab}$-enriched category, any initial object is also terminal and, more generally, finite products coincide with finite coproducts.

$\mathbf{Ab}$-enriched categories are well studied. For example:

• Preadditive categories are $\mathbf{Ab}$-enriched categories with a zero object (= initial and terminal)
• Additive categories are preadditive categories with finite biproducts;
• Abelian categories are additive categories with all kernels and cokernels, such that every monomorphism is a kernel and every epimorphism is a cokernel.

Answer 3

Other answers have argued, correctly, that there is in general no way to put a natural group structure on the set of morphisms between two objects.

To see that there is literally no way to do so, note that the set of morphisms from $X$ to $Y$ may be empty, and the empty set cannot be given an abelian group structure.