Editing Additive category (section)

== Internal characterisation of the addition law ==

Let '''C''' be a semiadditive category, so a category having all finitary biproducts. Then every hom-set has an addition, endowing it with the structure of an [[Abelian monoid#Commutative monoid|abelian monoid]], and such that the composition of morphisms is bilinear.

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

This shows that the addition law for an additive category is ''internal'' to that category.<ref>{{citation|mr=0049192|first=Saunders|last=MacLane|title=Duality for groups|journal=Bulletin of the American Mathematical Society|volume=56|issue=6|year=1950|pages=485–516
|url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183515045|doi=10.1090/S0002-9904-1950-09427-0|doi-access=free}} Sections 18 and 19 deal with the addition law in semiadditive categories.</ref>

To define the addition law, we will use the convention that for a biproduct, ''p<sub>k</sub>'' will denote the projection morphisms, and ''i<sub>k</sub>'' will denote the injection morphisms.

The ''diagonal morphism'' is the canonical morphism {{math|∆: ''A'' → ''A'' ⊕ ''A''}}, induced by the universal property of products, such that {{math|1=''p''<sub>''k''</sub> ∘ ∆ = 1<sub>''A''</sub>}} for {{math|1=''k'' = 1, 2}}. Dually, the ''codiagonal morphism'' is the canonical morphism {{math|∇: ''A'' ⊕ ''A'' → ''A''}}, induced by the universal property of coproducts, such that {{math|1=∇ ∘ ''i''<sub>''k''</sub> = 1<sub>''A''</sub>}} for {{math|1=''k'' = 1, 2}}.

For each object {{mvar|A}}, we define:

* the addition of the injections {{math|''i''<sub>1</sub> + ''i''<sub>2</sub>}} to be the diagonal morphism, that is {{math|1=∆ = ''i''<sub>1</sub> + ''i''<sub>2</sub>}};
* the addition of the projections {{math|''p''<sub>1</sub> + ''p''<sub>2</sub>}} to be the codiagonal morphism, that is {{math|1=∇ = ''p''<sub>1</sub> + ''p''<sub>2</sub>}}.

Next, given two morphisms {{math|α<sub>''k''</sub>: ''A'' → ''B''}}, there exists a unique morphism {{math|α<sub>1</sub> ⊕ α<sub>2</sub>: ''A'' ⊕ ''A'' → ''B'' ⊕ ''B''}} such that {{math|''p''<sub>''l''</sub> ∘ (α<sub>1</sub> ⊕ α<sub>2</sub>) ∘ ''i''<sub>''k''</sub>}} equals {{math|α<sub>''k''</sub>}} if {{math|1=''k'' = ''l''}}, and 0 otherwise.

We can therefore define {{math|1=α<sub>1</sub> + α<sub>2</sub> := ∇ ∘ (α<sub>1</sub> ⊕ α<sub>2</sub>) ∘ ∆}}.

This addition is both commutative and associative. The associativity can be seen by considering the composition
:<math>A\ \xrightarrow{\quad\Delta\quad}\ A \oplus A \oplus A\ \xrightarrow{\alpha_1\,\oplus\,\alpha_2\,\oplus\,\alpha_3}\ B \oplus B \oplus B\ \xrightarrow{\quad\nabla\quad}\ B</math>

We have {{math|1=α + 0 = α}}, using that {{math|1=α ⊕ 0 = ''i''<sub>1</sub> ∘ α ∘ ''p''<sub>1</sub>}}.

It is also bilinear, using for example that {{math|1=∆ ∘ β = (β ⊕ β) ∘ ∆}} and that {{math|1=(α<sub>1</sub> ⊕ α<sub>2</sub>) ∘ (β<sub>1</sub> ⊕ β<sub>2</sub>) = (α<sub>1</sub> ∘ β<sub>1</sub>) ⊕ (α<sub>2</sub> ∘ β<sub>2</sub>)}}.

We remark that for a biproduct {{math|''A'' ⊕ ''B''}} we have {{math|1=''i''<sub>1</sub> ∘ ''p''<sub>1</sub> + ''i''<sub>2</sub> ∘ ''p''<sub>2</sub> = 1}}. Using this, we can represent any morphism {{math|''A'' ⊕ ''B'' → ''C'' ⊕ ''D''}} as a matrix.