Additive category

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

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.


Definition[edit]

A category C is additive if

  1. it has a zero object
  2. 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
  3. 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.

Examples[edit]

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[edit]

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.[1]

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 ∆: AAA satisfying pk ∘ ∆ = 1A for k = 1, 2, and a
  • codiagonal morphism ∇: AAA satisfying ∇ ∘ ik = 1A for k = 1, 2.

Next, given two morphisms αk: AB, there exists a unique morphism α1 ⊕ α2: AABB 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

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 1 ⊕ α2) ∘ (β1 ⊕ β2) = (α1 ∘ β1) ⊕ (α2 ∘ β2).

We remark that for a biproduct AB we have i1 ∘ p1 + i2 ∘ p2 = 1. Using this, we can represent any morphism ABCD as a matrix.

Matrix representation of morphisms[edit]

Given objects A1, … , An and B1, … , Bm in an additive category, we can represent morphisms f: A1 ⊕ ⋅⋅⋅ ⊕ AnB1 ⊕ ⋅⋅⋅ ⊕ Bm 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 ik ∘ pk = 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 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.

If we interpret the object An 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[edit]

Recall that a functor F: CD 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.

Special cases[edit]

The additive categories most commonly studied are in fact abelian categories[citation needed]; for example, Ab is an abelian category.

References[edit]

  1. ^ 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  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