Additive category

In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A₁,...,A_n of C have a biproduct A₁ ⊕ ··· ⊕ A_n in C.

(Recall that a category C is preadditive if all its morphism sets are Abelian groups and morphism composition is bilinear, i.e. if C is enriched over the monoidal category of Abelian groups; and recall 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.

Contents

1 Examples 2 Elementary properties 3 Additive functors 4 Special cases 5 Sources

Examples

The original example of an additive category is the category Ab of Abelian groups with group homomorphisms. Ab is preadditive because it is a closed monoidal category, and the biproduct in Ab is the finite direct sum.

Other common examples:

• The category of (left) modules over a ring R, in particular:
  • the category of vector spaces over a field K.
• The algebra of matrices over a ring, thought of as a category as described below.

These will give you an idea of what to think of; for more examples, follow the links to Special cases below.

Elementary properties

Every additive category is of course a preadditive category, and many basic properties of these categories are described under that subject. This article concerns itself with the properties that exist specifically because of the existence of biproducts.

First note that because nullary biproducts exist, every additive category has a zero object, commonly denoted simply "0".

Given objects A and B in an additive category, we can use matrices to study the biproducts of A and B with themselves. Specifically, if we define the biproduct power Aⁿ to be the n-fold biproduct A ⊕ ··· ⊕ A and B^m similarly, then the morphisms from Aⁿ to B^m can be described as m-by-n matrices whose entries are morphisms from A to B.

For a concrete example, consider the category of real vector spaces, so that A and B are individual vector spaces. (There is no need for A and B to have finite dimensions, although of course the numbers m and n must be finite.) Then an element of Aⁿ may be represented as an n-by-Template:Num column vector whose entries are elements of A:

<math> \begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix} <math>

and a morphism from Aⁿ to B^m is an m-by-n matrix whose entries are morphisms from A to B:

<math> \begin{pmatrix} f_{1,1} & f_{1,2} & \cdots & f_{1,n} \\ f_{2,1} & f_{2,2} & \cdots & f_{2,n} \\ \vdots & \vdots & \cdots & \vdots \\ f_{m,1} & f_{m,2} & \cdots & f_{m,n} \end{pmatrix} <math>

Then this morphism matrix acts on the column vector by the usual rules of matrix multiplication to give an element of B^m, represented by an m-by-1 column vector with entries from B:

<math> \begin{pmatrix} f_{1,1} & f_{1,2} & \cdots & f_{1,n} \\ f_{2,1} & f_{2,2} & \cdots & f_{2,n} \\ \vdots & \vdots & \cdots & \vdots \\ f_{m,1} & f_{m,2} & \cdots & f_{m,n} \end{pmatrix} \begin{pmatrix} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{pmatrix} = \begin{pmatrix} f_{1,1}(a_{1}) + f_{1,2}(a_{2}) + \cdots + f_{1,n}(a_{n}) \\ f_{2,1}(a_{1}) + f_{2,2}(a_{2}) + \cdots + f_{2,n}(a_{n}) \\ \cdots \\ f_{m,1}(a_{1}) + f_{m,2}(a_{2}) + \cdots + f_{m,n}(a_{n}) \end{pmatrix} <math>

Even in the setting of an abstract additive category, where it makes no sense to speak of elements of the objects Aⁿ and B^m, the matrix representation of the morphism is still useful, because matrix multiplication correctly reproduces composition of morphisms. 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). 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. If we define morphism composition to be multiplication of matrices, then Mat(R) becomes an additive category, and A_n will be the biproduct power (A₁)ⁿ. In this way, matrices over a ring are seen to form an additive category, just as an individual ring formed a preadditive category (which in this case is End(A₁)). 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. However, this concept makes sense — such matrices have 0 entries are determined uniquely by their size alone — and while they are rather degenerate, they do need to be included to get an additive category, since an additive category must have a zero object 0. Thinking about such matrices can be useful in one way, however — they highlight the fact that given any objects A and B in an additive category, there is exactly one morphism from 0 to B (just as there is exactly one 1-by-0 matrix with entries in End(B)) and exactly one morphism from A to 0 (just as there is exactly one 0-by-1 matrix with entries in End(A)) -- 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. But if the categories are additive, then an additive functor can also be characterised as any functor that preserves biproduct diagrams. That is, if B is a biproduct of A₁,...,A_n in C with projection morphisms p_j and injection morphisms i_j, then F(B) should be a biproduct of F(A₁),...,F(A_n) in D with projection morphisms F(p_j) and injection morphisms F(i_j).

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, 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.

Sources

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