A category $\mathcal{C}$ is an ab-category or ab-category if
- for every pair of objects $A,B$ of $\mathcal{C}$ , there is a binary operation called addition, written $+_{(A,B)}$ or simply $+$ , defined on $\operatorname{hom}(A,B)$ ,
- the set $\hom(A,B)$ , together with $+$ is an abelian group,
- (left distributivity) if $f,g\in \hom(A,B)$ and $h\in \hom(B,C)$ , then $h(f+g)=hf+hg$ ,
- (right distributivity) if $f,g\in \hom(A,B)$ and $h\in \hom(C,A)$ , then $(f+g)h=fh+gh$ .
In a nutshell, an ab-category is a category in which every hom set in $\mathcal{C}$ is an abelian group such that morphism composition distributes over addition. Ab in the name stands for abelian, clearly indicative of the second condition above.
Since a group has a multiplicative (or additive if abelian) identity, $\hom(A,B)\neq\varnothing$ for every pair of objects $A,B$ in $\mathcal{C}$ . Furthermore, each $\hom(A,B)$ contains a unique morphism, written $0_{(A,B)}$ , as the additive identity of $\hom(A,B)$ . Because the subset $$\lbrace f\cdot 0_{(A,B)}\mid f\in\hom(B,C)\rbrace$$ of $\hom(A,C)$ is also a subgroup by right distributivity, and the additive identity of a subgroup coincides with the additive identity of the group, we have the following identity $$0_{(B,C)}0_{(A,B)}=0_{(A,C)}.$$
There are many examples of ab-categories, including the category of abelian groups, the category of $R$ -modules ($R$ a ring), the category of chain complexes, and the category of rings (not necessarily containing a multiplicative identity). However, the category of rings with 1 is not an ab-category (see below for more detail). Nevertheless, a unital ring $R$ itself considered as a category is an ab-category, as the ring of
endomorphisms clearly forms an abelian group. It is in fact a ring! This can be seen as a special case of the fact that, in an ab-category, $\operatorname{End}(A)=\hom(A,A)$ is always a ring (with 1). So, conversely, an ab-category with one object is a ring with 1, whose morphisms are elements of the ring.
If an ab-category has an initial object, that object is also a terminal object. By duality, the converse is also true. Therefore, in an ab-category, initial object, terminal object, and zero object are synonymous. In the category $\mathcal{R}$ of unital rings, $\mathbb{Z}$ is an initial object, but it has no terminal object, therefore $\mathcal{R}$ is not an ab-category.
An ab-category with a zero object $O$ is called a preadditive category.
In a preadditive category, the groups $\hom(A,O)$ and $\hom(O,B)$ are trivial groups by the definition of the zero object $O$ . Therefore, the zero morphism in $\hom(A,B)$ is also the additive identity of $\hom(A,B)$ : $$0_{(A,B)}=0_{(O,B)}0_{(A,O)}=A\longrightarrow O\longrightarrow B.$$
Most of the examples of ab-categories are readily seen to be preadditive. If a preadditive category $R$ has only one object, we see from above that it must be a ring. But this object must also be a zero object, so that $\operatorname{End}(R)$ must be trivial, which means $R$ itself must be trivial too, $R=0$ !
Remark. In some literature, a preadditive category is an ab-category, and some do not insist that a preadditive category contains a zero object. Here, we choose to differentiate the two.
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