A category C is preadditive if all its hom-sets are Abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of Abelian groups. A biproduct in a preadditive category is both a finitary product and a finitary coproduct.
A category C is additive if
- it has a zero object
- every hom-set Hom(A, B) has an addition, endowing it with the structure of an Abelian group, and such that composition of morphisms is bilinear
- all finitary biproducts exist.
Note that a category is called preadditive if just the second holds, whereas it is called semiadditive if both the first and the third hold.
Also, since the empty biproduct is a zero object in the category, we may omit the first condition. If we do this, however, we need to presuppose that the category C has zero morphisms, or equivalently that C is enriched over the category of pointed sets.
The original example of an additive category is the category of abelian groups Ab. The zero object is the trivial group, the addition of morphisms is given point-wise, and biproducts are given by direct sums.
The algebra of matrices over a ring, thought of as a category as described below, is also additive.
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.
Matrix representation of morphisms
Given objects A1, … , An and B1, … , Bm in an additive category, we can represent morphisms f: A1 ⊕ ⋅⋅⋅ ⊕ An → B1 ⊕ ⋅⋅⋅ ⊕ Bm as m-by-n matrices
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 the endomorphism ring End(A). If we denote the n-fold product of A with itself by An, then morphisms from An to Am are m-by-n matrices with entries from the ring End(A).
Conversely, given any ring R, we can form a category Mat(R) by taking objects An indexed by the set of natural numbers (including zero) and letting the hom-set of morphisms from An to Am be the set of m-by-n matrices over R, and where composition is given by matrix multiplication. Then Mat(R) is an additive category, and An equals the n-fold power (A1)n.
This construction should be compared with the result that a ring is a preadditive category with just one object, shown here.
This may be confusing in the special case where m or n is zero, because we usually don't think of matrices 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 objects A and B in an additive category, there is exactly one morphism from A to 0 (just as there is exactly one 0-by-1 matrix with entries in End(A)) and exactly one morphism from 0 to B (just as there is exactly one 1-by-0 matrix with entries in End(B)) – this is just what it means to say that 0 is a zero object. Furthermore, the zero morphism from A to B is the composition of these morphisms, as can be calculated by multiplying the degenerate matrices.
Recall that a functor F: C → D between preadditive categories is additive if it is an abelian group homomorphism on each hom-set in C. If the categories are additive, though, then a functor is additive if and only if it preserves all biproduct diagrams.
That is, if B is a biproduct of A1, … , An in C with projection morphisms pk and injection morphisms kj, then F(B) should be a biproduct of F(A1), … , F(An) in D with projection morphisms F(pk) and injection morphisms F(ik).
Almost all functors studied between additive categories are additive. In fact, it is a theorem that all adjoint functors between additive categories must be additive functors (see here), and most interesting functors studied in all of category theory are adjoints.
- A pre-abelian category is an additive category in which every morphism has a kernel and a cokernel.
- An abelian category is a pre-abelian category such that every monomorphism and epimorphism is normal.
The additive 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) goes over all of this very slowly