That is, an Ab-category C is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation.
More precisely, Ab is a closed monoidal category.
Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0.
Extending this analogy, the fact that composition is bilinear in general becomes the distributivity of multiplication over addition.
Focusing on a single object A in a preadditive category, these facts say that the endomorphism hom-set Hom(A,A) is a ring, if we define multiplication in the ring to be composition.
Indeed, given a ring R, we can define a preadditive category R to have a single object A, let Hom(A,A) be R, and let composition be ring multiplication.
Since R is an abelian group and multiplication in a ring is bilinear (distributive), this makes R a preadditive category.
Many concepts from ring theory, such as ideals, Jacobson radicals, and factor rings can be generalized in a straightforward manner to this setting.
Most functors studied between preadditive categories are additive.
of additive functors and all natural transformations between them is also preadditive.
The latter example leads to a generalization of modules over rings: If
Again, virtually all concepts from the theory of modules can be generalised to this setting.
in C has the structure of an R-module, and composition of morphisms is R-bilinear.
Any finite product in a preadditive category must also be a coproduct, and conversely.
In fact, finite products and coproducts in preadditive categories can be characterised by the following biproduct condition: This biproduct is often written A1 ⊕ ··· ⊕ An, borrowing the notation for the direct sum.
This is because the biproduct in well known preadditive categories like Ab is the direct sum.
However, although infinite direct sums make sense in some categories, like Ab, infinite biproducts do not make sense (see Category of abelian groups § Properties).
The biproduct condition in the case n = 0 simplifies drastically; B is a nullary biproduct if and only if the identity morphism of B is the zero morphism from B to itself, or equivalently if the hom-set Hom(B,B) is the trivial ring.
A preadditive category in which every biproduct exists (including a zero object) is called additive.
Further facts about biproducts that are mainly useful in the context of additive categories may be found under that subject.
Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense.
Unlike with products and coproducts, the kernel and cokernel of f are generally not equal in a preadditive category.
When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A.
However, in a general preadditive category there may exist morphisms without kernels and/or cokernels.
There is a convenient relationship between the kernel and cokernel and the abelian group structure on the hom-sets.
Given parallel morphisms f and g, the equaliser of f and g is just the kernel of g − f, if either exists, and the analogous fact is true for coequalisers.
The alternative term "difference kernel" for binary equalisers derives from this fact.
A preadditive category in which all biproducts, kernels, and cokernels exist is called pre-abelian.
Further facts about kernels and cokernels in preadditive categories that are mainly useful in the context of pre-abelian categories may be found under that subject.
Most of these special cases of preadditive categories have all been mentioned above, but they're gathered here for reference.