Additive category

In mathematics, specifically in category theory, an additive category is a preadditive category C such that all finite collections of objects A₁,...,A_n of C have a biproduct A₁ ⊕ ⋯ ⊕ A_n in C.

(Recall that 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. Recall also that a biproduct in a preadditive category is both a finite product and a finite coproduct.)

Warning: The term "additive category" is sometimes applied to any preadditive category, but Wikipedia does not follow this older practice.

1 Definition
2 Examples
3 Internal Characterisation of the Addition Law
4 Matrix Representation of Morphisms
5 Additive functors
6 Special cases
7 References

Definition

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 finite 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.

Examples

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.

More generally, every module category over a ring R is additive, and so in particular, the category of vector spaces over a field K is additive.

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 finite 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.^[1]

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

$A\ \xrightarrow{\quad\Delta\quad}\ A \oplus A \oplus A\ \xrightarrow{\alpha_1\,\oplus\,\alpha_2\,\oplus\,\alpha_3}\ B \oplus B \oplus B\ \xrightarrow{\quad\nabla\quad}\ B$

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, ..., A_n and B₁, ..., B_m in an additive category, we can represent morphisms f: A₁ ⊕ ⋅⋅⋅ ⊕ A_n → B₁ ⊕ ⋅⋅⋅ ⊕ B_m as m-by-n matrices

$\begin{pmatrix} f_{11} & f_{12} & \cdots & f_{1n} \\ f_{21} & f_{22} & \cdots & f_{2n} \\ \vdots & \vdots & \cdots & \vdots \\ f_{m1} & f_{m2} & \cdots & f_{mn} \end{pmatrix}$ where $f_{kl}�:= p_k\circ f \circ i_l\colon A_l\to B_k.$

Using that ∑_k i_k ∘ p_k = 1, it follows that addition and composition of matrices obey the usual rules for matrix addition and matrix multiplication.

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 A^m 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 A_n indexed by the set of natural numbers (including zero) and letting the hom-set of morphisms from A_n to A_m 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 A_n 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.

If we interpret the object A_n as the left module Rⁿ, then this matrix category becomes a subcategory of the category of left modules over R.

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.

Additive functors

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, ..., A_n in C with projection morphisms p_k and injection morphisms k_j, then F(B) should be a biproduct of F(A₁), ..., F(A_n) in D with projection morphisms F(p_k) and injection morphisms F(i_k).

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.

Special cases

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.

References

^ MacLane, Saunders (1950), "Duality for groups", Bulletin of the American Mathematical Society 56 (6): 485–516, doi:10.1090/S0002-9904-1950-09427-0, MR 0049192, http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183515045 Sections 18 and 19 deal with the addition law in semiadditive categories.

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