Additive categories

#category_theory/abelian_categories

As we noted, an $A b$ -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 an $A b$ -category which has a null object and a biproduct for each pair of objects.

But what is a null object and what are biproducts?

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 ${Hom}_{A} (a, b)$ .

Biproducts

Our main example categories, namely $Ab$ and $R -Mod$ , have additional properties beyond the group structure on the hom-sets. 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 $\times$ 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 $A \oplus B$ ) as if it's a product; e.g., by referring to projection maps $π_{1} : A \oplus B \to A$ and $π_{2} : A \oplus B \to B$ .

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

Biproducts

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

with arrows that satisfy the identities

p_{1} i_{1} = 1_{a}, p_{2} i_{2} = 1_{b}, i_{1} p_{1} + i_{2} p_{2} = 1_{c} .

Note that this definition is entirely "internal" in that it refers only to morphisms between the objects $a$ and $b$ , as opposed to the usual product $a \times b$ or coproduct $a ⊔ b$ , 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 $a \overset{p_{1}}{\leftarrow} c \overset{p_{2}}{\to} b$ 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 $b$ ).

Fortunately, we have the following result:

Products, coproducts, and biproducts

Two objects $a$ and $b$ in an $A b$ -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 $p_{1}$ and $p_{2}$ is a product of $a$ and $b$ , while the object $c$ with the morphisms $i_{1}$ and $i_{2}$ 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 $A b$ -category, $A$ , the object $c$ is usually written $c = a \oplus b$ . This defines a functor $\oplus : A \times A \to A$ , with $f_{1} \oplus f_{2}$ defined for morphisms $f_{1} : a \to a^{'}$ and $f_{2} : b \to b^{'}$ either by the equations

p_{j}^{'} (f_{1} \oplus f_{2}) = f_{j} p_{j}

(i.e., as defined for the product functor $\times$ ), or by the equations

(f_{1} \oplus f_{2}) i_{k} = i_{k}^{'} f_{k}

(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 $a \times b$ with the coproduct functor $a ⊔ b$ is a natural isomorphism.

Generalizing biproducts

We can iterate this process. Given $a_{1}, \dots, a_{n} \in A$ we can form a product $⨁_{j} a_{j}$ characterized (up to isomorphism) by the diagram

a_{j} \overset{i_{j}}{\to} ⨁_{j} a_{j} \overset{p_{k}}{\to} a_{k}

and the equations

i_{1} p_{1} + \dots + i_{n} p_{n} = 1, p_{k} i_{j} = δ_{k j},

where $δ_{k j} = 0$ if $k \neq j$ and $δ_{k k} = 1$ .

Moreover, for given objects $a_{i}, c_{j} \in A$ with $1 \leq i \leq n$ and $1 \leq j \leq m$ there is an isomorphism of abelian groups

{Hom}_{A} (⨁_{k} c_{k}, ⨁_{j} a_{j}) ≃ ⨁_{j, k} {Hom}_{C} (c_{k}, a_{j})

This implies that each morphism $f : ⨁_{k} c_{k} \to ⨁_{j} a_{j}$ is determined by the $n \times m$ matrix of its components $f_{k j} : p_{j} f i_{k} : c_{k} \to a_{j}$ . Composition of morphisms is then given by the usual matrix product of the matrices of components.

Suggested next note

Abelian categories