Additive categories

#category_theory/abelian_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 $s \to c$ . Dually, an object $t$ is terminal if for every object $c$ there exists a unique morphism $c \to t$ . 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:

In terms of universal properties, if we let $F : C \to Set$ be the functor that assigns to every object $c$ the singleton set $F (c) = {1}$ , and to every morphism $f : c \to c^{'}$ the identity set map $F (f) : {1} \to {1}$ , then an initial object $s$ is characterized by a natural bijection
$ϕ_{c} : {Hom}_{C} (s, c) \tilde{\to} F (c),$
while a terminal object is characterized by a natural bijection
$ψ_{c} : {Hom}_{C} (c, t) \tilde{\to} F (c) .$
In terms of limits and colimits, an initial object of $C$ is a limit of the empty diagram in $C$ , while a terminal object is a colimit of the empty diagram in $C$ .
In terms of adjoints, we have the following:

If $C \to 1$ is the unique functor to the category with one object, then a left adjoint $1 \to C$ is the functor that selects the initial object of $C$ (assuming it exists), while a right adjoint $1 \to C$ is the functor that selects the terminal object of $C$ (assuming it exists).

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

Examples

In $Set$ , the empty set is an initial object and any singleton set is a terminal object. For each set $X$ , the unique set map $\emptyset \to X$ is the empty map, while the unique set map $X \mapsto {*}$ is the map $x \mapsto *$ . There is no null object.
In $A b$ , the trivial group is a null object. For each abelian group $G$ , the unique group morphism ${0} \to A$ is the map $0 \mapsto 0_{A}$ , while the unique group morphism $A \to {0}$ is the trivial map $a \mapsto 0$ .
In $Grp$ , the trivial group is a null object. For each group $G$ , the unique group morphism ${e} \to G$ is the map $e \mapsto e_{G}$ , while the unique group morphism $G \to {e}$ is the trivial map $g \mapsto e$ .
In $Ring$ , the ring of integers $Z$ is an initial object. For each ring $R$ (with unity), the unique ring morphism $Z \to R$ is determined entirely by $1_{Z} \mapsto 1_{R}$ . If we allow rings with $0 = 1$ , then the zero ring is a terminal object. (There are no ring morphisms from the zero ring to any other ring!)
In $Fld$ , there is neither an initial nor a terminal object. In the category of fields of characteristic $0$ , the field of rational numbers $Q$ is an initial object (but there is no terminal object). In the category of fields of fixed characteristic $p > 0$ , the field $F_{p} ≃ Z_{p}$ is initial (but there is no terminal object).
In $Cat$ , the empty category $0$ is initial and the category $1$ is terminal.
A limit of a diagram $F$ is a terminal object in the category of cones to $F$ . A colimit of $F$ is an initial object in the category of cones from $F$ .

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 $a \to b$ that factors through the unique morphisms to and from $z$ :

a \to z \to b .

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

In $A b$ , the zero morphism $A \to B$ between two abelian groups is the trivial map $a \mapsto 0_{B}$ . More generally, when $C$ is an additive category the zero morphism $a : \to b$ is the additive identity of the abelian group ${Hom}_{C} (a, b)$ . In $G r p$ , the zero morphism $G \to H$ between two groups is the trivial map $g \mapsto e_{H}$ .

One can show (are you the one?) that in an additive category, each zero morphism $0 : a \to b$ is (as hoped!) the additive identity of the abelian group ${Hom}_{A} (a, b)$ .

Suggested next note

Abelian categories