Inmathematics, specifically incategory theory, anadditive category is apreadditive category C admitting allfinitarybiproducts.

There are two equivalent definitions of an additive category: One as acategory equipped with additional structure, and another as a category equipped withno extra structure but whose objects andmorphisms satisfy certain equations.

A categoryC is preadditive if all itshom-sets areabelian groups and composition of morphisms isbilinear; in other words,C isenriched over themonoidal category of abelian groups.

In a preadditive category, every finitaryproduct is necessarily acoproduct, and hence abiproduct, andconversely every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it). The empty product, is afinal object and the empty product in the case of an empty diagram, aninitial object. Both being limits, they are not finite products nor coproducts.

Thus an additive category is equivalently described as a preadditive category admitting all finitary products and with the null object or a preadditive category admitting all finitary coproducts and with the null object

We give an alternative definition.

Define asemiadditivecategory to be a category (note: not a preadditive category) which admits azero object and all binarybiproducts. It is then a remarkable theorem that the Hom sets naturally admit anabelian monoid structure. Aproof of this fact is given below.

An additive category may then be defined as a semiadditive category in which every morphism has anadditive inverse. This then gives the Hom sets anabelian group structure instead of merely an abelian monoid structure.

More generally, one also considers additiveR-linear categories for acommutative ringR. These are categories enriched over the monoidal category ofR-modules and admitting all finitary biproducts.

The original example of an additive category is thecategory of abelian groupsAb. The zero object is thetrivial group, the addition of morphisms is givenpointwise, and biproducts are given bydirect sums.

More generally, everymodule category over aringR is additive, and so in particular, thecategory of vector spaces over afieldK is additive.

The algebra ofmatrices over a ring, thought of as a category as described below, is also additive.

LetC be a semiadditive category, so a category having all finitary biproducts. Then every hom-set has an addition, endowing it with the structure of anabelian monoid, and such that the composition of morphisms is bilinear.

Moreover, ifC is additive, then the two additions on hom-sets must agree. In particular, a semiadditive category is additiveif and only if every morphism has an additive inverse.

This shows that the addition law for an additive category isinternal to that category.^[1]

To define the addition law, we will use the convention that for a biproduct,p_k will denote the projection morphisms, andi_k will denote the injection morphisms.

Thediagonal morphism is the canonical morphism∆:A →A ⊕A, induced by the universal property of products, such thatp_k ∘ ∆ = 1_A fork = 1, 2. Dually, thecodiagonal morphism is the canonical morphism∇:A ⊕A →A, induced by the universal property of coproducts, such that∇ ∘ i_k = 1_A fork = 1, 2.

For each objectA, we define:

• the addition of the injectionsi₁ +i₂ to be the diagonal morphism, that is∆ =i₁ +i₂;
• the addition of the projectionsp₁ +p₂ to be the codiagonal morphism, that is∇ =p₁ +p₂.

Next, given two morphismsα_k:A →B, there exists a unique morphismα₁ ⊕ α₂:A ⊕A →B ⊕B such thatp_l ∘ (α₁ ⊕ α₂) ∘i_k equalsα_k ifk =l, and 0 otherwise.

We can therefore defineα₁ + α₂ := ∇ ∘ (α₁ ⊕ α₂) ∘ ∆.

This addition is both commutative and associative. The associativity can be seen by considering the composition

${\displaystyle A\stackrel{\mathrm{\Delta }}{\to }A\oplus A\oplus A\stackrel{{\alpha }_1\oplus {\alpha }_2\oplus {\alpha }_3}{\to }B\oplus B\oplus B\stackrel{\mathrm{\nabla }}{\to }B}$

We haveα + 0 = α, using thatα ⊕ 0 =i₁ ∘ α ∘ p₁.

It is also bilinear, using for example that∆ ∘ β = (β ⊕ β) ∘ ∆ and that(α₁ ⊕ α₂) ∘ (β₁ ⊕ β₂) = (α₁ ∘ β₁) ⊕ (α₂ ∘ β₂).

We remark that for a biproductA ⊕B we havei₁ ∘ p₁ +i₂ ∘ p₂ = 1. Using this, we can represent any morphismA ⊕B →C ⊕D as a matrix.

Given objectsA₁, ..., A_n andB₁, ..., B_m in an additive category, we can represent morphismsf:A₁ ⊕ ⋅⋅⋅ ⊕A_n →B₁ ⊕ ⋅⋅⋅ ⊕B_m asm-by-n matrices

where${\displaystyle f_{kl}:=p_k\circ f\circ i_l:A_l\to B_k.}$

Using that∑_ki_k ∘ p_k = 1, it follows that addition and composition of matrices obey the usual rules formatrix addition andmultiplication.

Thus additive categories can be seen as the most general context in which the algebra of matrices makes sense.

Recall that the morphisms from a single object A to itself form theendomorphism ringEnd A.If we denote then-fold product of A with itself byAⁿ, then morphisms fromAⁿ toA^m arem-by-n matrices with entries from the ringEnd A.

Conversely, given anyringR, we can form a category Mat(R) by taking objectsA_n indexed by the set ofnatural numbers (including0) and letting the hom-set of morphisms fromA_n toA_m be theset ofm-by-n matrices over R, and where composition is given by matrix multiplication.^[2] ThenMat(R) is an additive category, andA_n equals then-fold power(A₁)ⁿ.

This construction should be compared with the result that a ring is a preadditive category with just one object, shownhere.

If we interpret the objectA_n as the leftmodule Rⁿ, then thismatrix category becomes asubcategory of the category of left modules over R.

This may be confusing in the special case wherem orn is zero, because we usually don't think ofmatrices with 0 rows or 0 columns. This concept makes sense, however: such matrices have no entries and so are completely determined by their size. While these matrices are rather degenerate, they do need to be included to get an additive category, since an additive category must have a zero object.

Thinking about such matrices can be useful in one way, though: they highlight the fact that given any objectsA andB in an additive category, there is exactly one morphism fromA to 0 (just as there is exactly one 0-by-1 matrix with entries inEnd A) and exactly one morphism from 0 toB (just as there is exactly one 1-by-0 matrix with entries inEnd B) – this is just what it means to say that 0 is azero object. Furthermore, the zero morphism fromA toB is the composition of these morphisms, as can be calculated by multiplying the degenerate matrices.

AfunctorF:C →D between preadditive categories isadditive if it is an abelian grouphomomorphism on eachhom-set inC. If the categories are additive, then a functor is additive if and only if it preserves allbiproduct diagrams.

That is, ifB is a biproduct of A₁, ... , A_n in C with projection morphismsp_k and injection morphismsi_j, thenF(B) should be a biproduct of F(A₁), ... , F(A_n) in D with projection morphismsF(p_j) and injection morphismsF(i_j).

Almost all functors studied between additive categories are additive. In fact, it is a theorem that alladjoint functors between additive categories must be additive functors (seehere). Most of the interesting functors studied in category theory are adjoints.

When considering functors betweenR-linear additive categories, one usually restricts toR-linear functors, so those functors giving anR-module homomorphism on each hom-set.

• Apre-abelian category is an additive category in which every morphism has akernel and acokernel.
• Anabelian category is a pre-abelian category such that everymonomorphism andepimorphism isnormal.

Many commonly studied additive categories are in fact abelian categories; for example,Ab is an abelian category. Thefree abelian groups provide an example of a category that is additive but not abelian.^[3]

• Nicolae Popescu; 1973;Abelian Categories with Applications to Rings and Modules; Academic Press, Inc. (out of print) goes over all of this very slowly

• Additive categories

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