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Inmathematics, specifically incategory theory, apreadditive category is another name for anAb-category, i.e., acategory that isenriched over thecategory of abelian groups,Ab.That is, anAb-categoryC is acategory such thateveryhom-set Hom(A,B) inC has the structure of an abelian group, and composition of morphisms isbilinear, in the sense that composition of morphisms distributes over the group operation.In formulas:$${\displaystyle f\circ (g+h)=(f\circ g)+(f\circ h)}$$and$${\displaystyle (f+g)\circ h=(f\circ h)+(g\circ h),}$$where + is the group operation.

Some authors have used the termadditive category for preadditive categories, but this page reserves that term for certain special preadditive categories (see§ Special cases below).

The most obvious example of a preadditive category is the categoryAb itself. More precisely,Ab is aclosed monoidal category. Note thatcommutativity is crucial here; it ensures that the sum of twogroup homomorphisms is again a homomorphism. In contrast, the category of allgroups is not closed. SeeMedial category.

Other common examples:

• The category of (left)modules over aringR, in particular:
  • thecategory of vector spaces over afieldK.
• The algebra ofmatrices over a ring, thought of as a category as described in the articleAdditive 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§ Special cases.

Because every hom-set Hom(A,B) is an abelian group, it has azero element 0. This is thezero morphism fromA toB. 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 thedistributivity of multiplication over addition.

Focusing on a single objectA in a preadditive category, these facts say that theendomorphism hom-set Hom(A,A) is aring, if we define multiplication in the ring to be composition. This ring is theendomorphism ring ofA. Conversely, every ring (withidentity) is the endomorphism ring of some object in some preadditive category. Indeed, given a ringR, we can define a preadditive categoryR to have a single objectA, let Hom(A,A) beR, and let composition be ring multiplication. SinceR is an abelian group and multiplication in a ring is bilinear (distributive), this makesR a preadditive category. Category theorists will often think of the ringR and the categoryR as two different representations of the same thing, so that a particularlyperverse category theorist might define a ring as a preadditive category with exactlyone object (in the same way that amonoid 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 asideals,Jacobson radicals, andfactor 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".

If${\displaystyle C}$ and${\displaystyle D}$ are preadditive categories, then afunctor${\displaystyle F:C\to D}$ isadditive if it too isenriched over the category${\displaystyle Ab}$. That is,${\displaystyle F}$ is additiveif and only if, given any objects${\displaystyle A}$ and${\displaystyle B}$ of${\displaystyle C}$, thefunction${\displaystyle F:\text{Hom}(A,B)\to \text{Hom}(F(A),F(B))}$ is agroup homomorphism. Most functors studied between preadditive categories are additive.

For a simple example, if the rings${\displaystyle R}$ and${\displaystyle S}$ are represented by the one-object preadditive categories${\displaystyle C_R}$ and${\displaystyle C_S}$, then aring homomorphism from${\displaystyle R}$ to${\displaystyle S}$ is represented by an additive functor from${\displaystyle C_R}$ to${\displaystyle C_S}$, and conversely.

If${\displaystyle C}$ and${\displaystyle D}$ are categories and${\displaystyle D}$ is preadditive, then thefunctor category${\displaystyle D^C}$ is also preadditive, becausenatural transformations can be added in a natural way.If${\displaystyle C}$ is preadditive too, then the category${\displaystyle \text{Add}(C,D)}$ of additive functors and all natural transformations between them is also preadditive.

The latter example leads to a generalization ofmodules over rings: If${\displaystyle C}$ is a preadditive category, then${\displaystyle \text{Mod}(C):=\text{Add}(C,Ab)}$ is called themodule category over${\displaystyle C}$.^{[citation needed]} When${\displaystyle C}$ is the one-object preadditive category corresponding to the ring${\displaystyle R}$, this reduces to the ordinary category of(left)${\displaystyle R}$-modules. Again, virtually all concepts from the theory of modules can be generalised to this setting.

More generally, one can consider a categoryC enriched over the monoidal category ofmodules over acommutative ringR, called anR-linear category. In other words, eachhom-set${\displaystyle \text{Hom}(A,B)}$ inC has the structure of anR-module, and composition of morphisms isR-bilinear.

When considering functors between twoR-linear categories, one often restricts to those that areR-linear, so those that induceR-linear maps on each hom-set.

Anyfiniteproduct in a preadditive category must also be acoproduct, and conversely. In fact, finite products and coproducts in preadditive categories can be characterised by the followingbiproduct condition:

The objectB is abiproduct of the objectsA₁, ...,A_nif and only if there areprojection morphismsp_j: B → A_j andinjection morphismsi_j: A_j → B, such that (i₁∘p₁) + ··· + (i_n∘p_n) is the identity morphism ofB,p_j∘i_j is theidentity morphism ofA_j, andp_j∘i_k is the zero morphism fromA_k toA_j wheneverj andk aredistinct.

This biproduct is often writtenA₁ ⊕ ··· ⊕ A_n, borrowing the notation for thedirect sum. This is because the biproduct in well known preadditive categories likeAbis the direct sum. However, althoughinfinite direct sums make sense in some categories, likeAb, infinite biproducts donot make sense (seeCategory of abelian groups § Properties).

The biproduct condition in the casen = 0 simplifies drastically;B is anullary biproduct if and only if the identity morphism ofB is the zero morphism fromB to itself, or equivalently if the hom-set Hom(B,B) is thetrivial ring. Note that because a nullary biproduct will be bothterminal (a nullary product) andinitial (a nullary coproduct), it will in fact be azero object.Indeed, the term "zero object" originated in the study of preadditive categories likeAb, where the zero object is thezero group.

A preadditive category in which every biproduct exists (including a zero object) is calledadditive. 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 ofkernel andcokernelmake sense. That is, iff: A → B is amorphism in a preadditive category, then the kernel off is theequaliser off and the zero morphism fromA toB, while the cokernel off is thecoequaliser off and this zero morphism. Unlike with products and coproducts, the kernel and cokernel off 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 akernel of a homomorphism, if one identifies the ordinary kernelK off: A → B with its embeddingK → 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 morphismsf andg, the equaliser off andg is just the kernel ofg − 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 calledpre-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.

• Aring is a preadditive category with exactly one object.
• Anadditive category is a preadditive category with all finite biproducts.
• Apre-abelian category is an additive category with all kernels and cokernels.
• Anabelian category is a pre-abelian category such that everymonomorphism andepimorphism isnormal.

The preadditive categories most commonly studied are in fact abelian categories; for example,Ab is an abelian category.

• 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

• Additive categories

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