Ab-categories

Ab-categories

There are many familiar categories in which the hom-sets have additional structure:

We are going to focus on the examples of the categories Ab and R-Mod, in which the hom-sets have the structure of abelian groups.

Definition of Ab-category

An Ab-category (also called a preadditive category) is a category A in which each hom-set HomA(a,b) is an abelian group and for which the composition is bilinear. In other words, for morphisms f,f:ab and g,g:bc we have

(g+g)(f+f)=gf+gf+gf+gf.

Note that because the composition of morphisms is bilinear, it can also be written using the tensor product (over Z) as a linear map:

HomA(b,c)ZHomA(a,b)HomA(a,c)

It is possible to define an Ab-category directly (without first defining a category), as given by the following data:

  1. A set of objects.
  2. A function that assigns to each ordered pair of objects (b,c) an abelian group A(b,c).
  3. For each ordered triple of objects (a,b,c) a morphism of abelian groupsA(b,c)ZA(a,b)A(a,c).This morphism is called "composition" and written gfgf.
  4. For each object a a group morphism ZA(a,a). (This is the analogue of each object in a category having a unique identity arrow, which corresponds to a set map {}HomA(a,a).)
    These data are required to satisfy the usual associative and unit laws for composition. This is a definition of Ab-category completely analogous to the definition of a conventional category, with:

If we are dealing with Ab-categories, we will probably want to restrict our functors to those that respect that morphism addition:

Definition of additive functors

If A and B are Ab-categories, a functor T:AB is said to be additive when every function T:HomA(a,a)HomB(T(a),T(a)) is a group morphism; i.e., when T(f+f)=T(f)+T(f) for all parallel morphisms f,f.

Biproducts

Our main example categories, namely Ab and R-Mod have additional properties beyond the hom-sets being abelian groups. One of those properties is that products and coproducts always exist for every pair of objects, and they're always isomorphic. In other words, the product functor × and coproduct functor are naturally isomorphic. Because of this, there is usually some confusion as to which type of product to use, and you often see textbooks using coproduct (usually called the direct sum and denoted AB) as if it's a product; e.g., by referring to projection maps π1:ABA and π2:ABB.

This situation can happen more generally in any Ab-category. We first introduce a new type of product:

Biproducts

In a Ab-category A, a biproduct diagram for a pair of objects a,b is a diagram

with arrows that satisfy the identities

p1i1=1a,p2i2=1b,i1p1+i2p2=1c.

Note that this definition is entirely "internal" in that it refers only to morphisms to and from the objects a and b, as opposed to the usual product a×b or coproduct ab, which are defined as limits and colimits and hence involve morphisms to and from every object in the category.

In other words, to verify you have a biproduct diagram, you just need to directly verify the three identity relations above. On the other hand, to verify ap1cp2b is a product you need to verify it satisfies the appropriate universal property, which involves comparing c (with its projections) to every other object d (that also has projections to a and b).

Products, coproducts, and biproducts

Two objects a and b in an Ab-category A have a product in A if and only if they have a biproduct in A. In the biproduct diagram above, the object c with the morphisms p1 and p2 is a product of a and b, while the object c with the morphisms i1 and i2 is a coproduct of a and b.

In particular, two objects a and b have a product in A exactly when they have a coproduct in A.

In the categories Ab and R-Mod the biproduct is usually called the direct sum of the given objects. Because of this, if the biproduct diagram exists for all a,b in a given Ab-category A, the object c is usually written c=ab. This defines a functor :A×AA, with f1f2 defined for morphisms f1:aa and f2:bb either by the equations

pj(f1f2)=fjpj

(i.e., as defined for the product functor ×), or by the equations

(f1f2)ik=ikfk

(i.e., as defined for the coproduct functor ). By the properties of the biproduct diagram, either choice implies the other.

In other words, the identification of the product functor a×b with the coproduct functor ab is a natural isomorphism.


We can iterate this process. Given a1,,anA we can form a product jaj characterized (up to isomorphism) by the diagram

ajijjajpkak

and the equations

i1p1++inpn=1,pkij=δkj,

where δkj=0 if kj and δkk=1.

Moreover, for given objects ai,cjA with 1in and 1jm there is an isomorphism of abelian groups

HomA(kck,jaj)j,kHomC(ck,aj)

This implies that each morphism f:kckjaj is determined by the n×m matrix of its components fkj:pjfik:ckaj. Composition of morphisms is then given by the usual matrix product of the matrices of components.