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:
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
As usual, there are various equivalent interpretations of these properties:
-
In terms of universal properties, if we let
be the functor that assigns to every object the singleton set , and to every morphism the identity set map , then an initial object is characterized by a natural bijection while a terminal object is characterized by a natural bijection
-
In terms of limits and colimits, an initial object of
is a limit of the empty diagram in , while a terminal object is a colimit of the empty diagram in . -
In terms of adjoints, we have the following:
In any case, when they exist, initial and terminal objects are unique up to unique isomorphism. The same goes for null objects.
Examples
- In
, the empty set is an initial object and any singleton set is a terminal object. For each set , the unique set map is the empty map, while the unique set map is the map . There is no null object. - In
, the trivial group is a null object. For each abelian group , the unique group morphism is the map , while the unique group morphism is the trivial map . - In
, the trivial group is a null object. For each group , the unique group morphism is the map , while the unique group morphism is the trivial map . - In
, the ring of integers is an initial object. For each ring (with unity), the unique ring morphism is determined entirely by . If we allow rings with , then the zero ring is a terminal object. (There are no ring morphisms from the zero ring to any other ring!) - In
, there is neither an initial nor a terminal object. In the category of fields of characteristic , the field of rational numbers is an initial object (but there is no terminal object). In the category of fields of fixed characteristic , the field is initial (but there is no terminal object). - In
, the empty category is initial and the category is terminal. - A limit of a diagram
is a terminal object in the category of cones to . A colimit of is an initial object in the category of cones from .
Zero morphisms
If a category
This morphism is called the zero morphism between
In
One can show (are you the one?) that in an additive category, each zero morphism