Additive categories

As we noted, a preadditive category is sometimes also called a preadditive category, suggesting that there is something called an additive category. Indeed:

Definition of an additive category

An additive category is a preadditive category which has a null object and a biproduct for each pair of objects.

If you're familiar with null objects, feel free to move along to the next note. If you would like a primer, keep reading.

Null objects

Recall that in a fixed category C, a object s is initial if for every object c there exists a unique morphism s→c. Dually, an object t is terminal if for every object c there exists a unique morphism c→t. An object that is both initial and terminal is called a null object (or a zero object).

As usual, there are various equivalent interpretations of these properties:

If C→1 is the unique functor to the category with one object, then a left adjoint 1→C is the functor that selects the initial object of C, while a right adjoint 1→C is the functor that selects the terminal object of C.

In any case, when they exist, initial and terminal objects are unique up to unique isomorphism. The same goes for null objects.

Examples

Zero morphisms

If a category C has a null object z, then for every pair of objects a and b in C there is a unique morphism a→b that factors through the unique morphisms to and from z:

a→z→b.

This morphism is called the zero morphism between a and b and is denoted 0:a→b. Any composite with a zero morphism is another zero morphism.

In Ab, the zero morphism Aβ†’B between two abelian groups is the trivial map a↦0B. More generally, when C is an additive category the zero morphism a:β†’b is the additive identity of the abelian group HomC(a,b). In Grp, the zero morphism Gβ†’H between two groups is the trivial map g↦eH.

One can show (are you the one?) that in an additive category, each zero morphism 0:a→b is (as hoped!) the additive identity of the abelian group HomA(a,b).


Suggested next note

Abelian categories