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, pk will denote the projection morphisms, and ik will denote the injection morphisms.
We first observe that for each object A there is a
- diagonal morphism ∆: A → A ⊕ A satisfying pk ∘ ∆ = 1A for k = 1, 2, and a
- codiagonal morphism ∇: A ⊕ A → A satisfying ∇ ∘ ik = 1A for k = 1, 2.
Next, given two morphisms αk: A → B, there exists a unique morphism α1 ⊕ α2: A ⊕ A → B ⊕ B such that pl ∘ (α1 ⊕ α2) ∘ ik equals αk 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.
Read more about this topic: Additive Category
Famous quotes containing the words internal, addition and/or law:
“We all run on two clocks. One is the outside clock, which ticks away our decades and brings us ceaselessly to the dry season. The other is the inside clock, where you are your own timekeeper and determine your own chronology, your own internal weather and your own rate of living. Sometimes the inner clock runs itself out long before the outer one, and you see a dead man going through the motions of living.”
—Max Lerner (b. 1902)
“The force of truth that a statement imparts, then, its prominence among the hordes of recorded observations that I may optionally apply to my own life, depends, in addition to the sense that it is argumentatively defensible, on the sense that someone like me, and someone I like, whose voice is audible and who is at least notionally in the same room with me, does or can possibly hold it to be compellingly true.”
—Nicholson Baker (b. 1957)
“If he who breaks the law is not punished, he who obeys it is cheated. This, and this alone, is why lawbreakers ought to be punished: to authenticate as good, and to encourage as useful, law-abiding behavior. The aim of criminal law cannot be correction or deterrence; it can only be the maintenance of the legal order.”
—Thomas Szasz (b. 1920)