In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups. That is, an Ab-category is a category such that every hom-set has the structure of an abelian group, and composition of morphisms is bilinear is the group operation.

Some authors have used the term additive category for preadditive categories, but this article reserves that term for certain special preadditive categories (see below).

Examples

is a closed monoidal category. Note that commutativity is crucial here; it ensures that the sum of two group homomorphisms is again a homomorphism. In contrast, the category of all groupss is not closed. See Medial category.

Other common examples:

The category of (left) moduless over a ring, in particular:

• the category of vector spaces over a field.

• The algebra of matrices over a ring, thought of as a category as described in the article Additive category.

• Any ring, thought of as a category with only one object, is a preadditive category. Here composition of morphisms is just ring multiplication and the unique hom-set is the underlying abelian group.

For more examples, see .

Elementary properties

is an abelian group, it has a zero element 0. This is the zero morphism. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in general becomes the distributivity of multiplication over addition.

in a preadditive category, these facts say that the endomorphism is a ring, if we define multiplication in the ring to be composition. This ring is the endomorphism ring. Conversely, every ring (with identity as two different representations of the same thing, so that a particularly perverse category theorist might define a ring as a preadditive category with exactly one object (in the same way that a monoid can be viewed as a category with only one object—and forgetting the additive structure of the ring gives us a monoid).

In this way, preadditive categories can be seen as a generalisation of rings. Many concepts from ring theory, such as idealss, Jacobson radicals, and factor rings can be generalized in a straightforward manner to this setting. When attempting to write down these generalizations, one should think of the morphisms in the preadditive category as the "elements" of the "generalized ring".

Additive functors

are preadditive categories, then a functor is additive if it too is enriched is additive if and only if, the function is a group homomorphism. Most functors studied between preadditive categories are additive.

, then a ring homomorphism, and conversely.

is preadditive, then the functor category is also preadditive, because natural transformations of additive functors and all natural transformations between them is also preadditive.

The latter example leads to a generalization of modules is called the module category., this reduces to the ordinary category of R-modules. Again, virtually all concepts from the theory of modules can be generalised to this setting.

R-linear categories

enriched over the monoidal category of modules over a commutative ring, called an -linear category. In other words, each hom-set-bilinear.

-linear maps on each hom-set.

Biproducts

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:

1. is a biproduct if and only if there are projection morphisms and injection morphisms is the identity morphism are distinct.

, borrowing the notation for the direct sum is the direct sum. However, although infinite, infinite biproducts do not make sense (see ).

is a nullary biproduct is the trivial ring. Note that because a nullary biproduct will be both terminal (a nullary product) and initial (a nullary coproduct), it will in fact be a zero object, where the zero object is the zero group.

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.

Kernels and cokernels

Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel is the equaliser is the coequaliser 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. However, in a general preadditive category there may exist morphisms without kernels and/or cokernels.

, 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.

Special cases

Most of these special cases of preadditive categories have all been mentioned above, but they're gathered here for reference.

• A ring is a preadditive category with exactly one object.

• An additive category is a preadditive category with all finite biproducts.

• A pre-abelian category is an additive category with all kernels and cokernels.

• An abelian category is a pre-abelian category such that every monomorphism and epimorphism is normal.

is an abelian category.

References

• Nicolae Popescu; 1973; Abelian Categories with Applications to Rings and Modules; Academic Press, Inc.; out of print

• Charles Weibel; 1994; An introduction to homological algebra; Cambridge Univ. Press