Additive Categories - Internal Characterisation of The Addition Law

Internal Characterisation of The Addition Law

Let C be a semiadditive category, so a category having

• a zero object
• all finitary biproducts.

Then every hom-set has an addition, endowing it with the structure of an 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.

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

We first observe that for each object A there is a

• diagonal morphism ∆: A → A ⊕ A satisfying p_k ∘ ∆ = 1_A for k = 1, 2, and a
• codiagonal morphism ∇: A ⊕ A → A satisfying ∇ ∘ i_k = 1_A for k = 1, 2.

Next, given two morphisms α_k: A → B, there exists a unique morphism α₁ ⊕ α₂: A ⊕ A → B ⊕ B such that p_l ∘ (α₁ ⊕ α₂) ∘ i_k equals α_k if k = l, and 0 otherwise.

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

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

We have α + 0 = α, using that α ⊕ 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 biproduct A ⊕ B we have i₁ ∘ p₁ + i₂ ∘ p₂ = 1. Using this, we can represent any morphism A ⊕ B → C ⊕ D as a matrix.