# I've read that abelian categories can naturally be enriched in $\mathbf{Ab}.$ How does this actually work?

Wikipedia defines the notion of an abelian category as follows (link).

A category is abelian iff

1. it has a zero object,
2. it has all binary products and binary coproducts, and
3. it has all kernels and cokernels.
4. all monomorphisms and epimorphisms are normal.

It later explains that an abelian category can naturally be enriched in $\mathbf{Ab},$ as a result of the first three axioms above.

How does this actually work?

• Odd, because the version of Abelian category I learned is precisely that the Hom-sets are Abelian groups :p – Malice Vidrine Jan 3 '14 at 3:18
• @user18921 You need the fourth axiom. $\mathbf{Grp}$ satisfies the first three axioms but is not enriched in $\mathbf{Ab}$. – Zhen Lin Jan 3 '14 at 3:43
• General hint: Don't learn mathematics through Wikipedia. Take books for this. How abelian categories behave etc. is treated in many introductions to category theory, for example Saunders Mac Lane's classic CWM. – Martin Brandenburg Jan 3 '14 at 11:29

The claim is not trivial and requires a bit of ingenuity. For a guided solution, see Q6 here. The four steps are as follows:

1. Show that the category has finite limits and finite colimits.
2. Show that a morphism $f$ is monic (resp. epic) if and only if $\ker f = 0$ (resp. $\operatorname{coker} f = 0$).
3. Show that the canonical morphisms $A + B \to A \times B$ are isomorphisms, and then deduce that the category is enriched over commutative monoids.
4. Show that every morphism has an additive inverse (by inverting an appropriate matrix).

A complete proof is given as Theorem 1.6.4 in [Borceux, Handbook of categorical algebra, Vol. 2].

• What do you mean by the following statement? "...and then deduce that the category is enriched over commutative monoids." – goblin Jan 3 '14 at 6:22
• It means every hom-set has a commutative monoid structure, and composition is bilinear. – Zhen Lin Jan 3 '14 at 11:28

I would leave this as a comment, but it seems that I don't have enough reputation to do that, so…

I believe it comes down to showing that finite products are coproducts in additive categories, i.e. categories satisfying the first three of your axioms (in light of the first lemma here, for example). This has been answered, for example, here:

In an additive category, why is finite products the same as finite coproducts?