Additive categories

Initial objects, terminal objects, and null objects

Definition of initial and terminal objects

In a fixed category $C$ , and 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:

Saunders Mac Lane

...good general theory does not search for the maximum generality, but for the right generality.

Motivation

Have you ever wondered what's so "free" about free groups? Or why the free abelian group on a set $X$ is not the same as the free group on the set $X$ ? Have you ever wondered what some constructions (such as these "free" constructions) commute with certain constructions (like forming direct sums) but not others (like forming direct products)?

Or perhaps you've wondered if objects in different categories can have maps to each other. Sadly, they cannot. But they can try the next best thing. Imagine for a moment, two categories $C$ and $D$ and objects $c \in C$ and $d \in D$ . Officially, the two objects are separated by an impossible gulf:

If there are no functors between $C$ and $D$ , then the objects $c$ and $d$ truly will never see each other. They can have no interaction whatsoever. The story ends.

But suppose there were functors connecting the two categories. We can think of these as transit lines, able to convey data (objects and arrows) from one category to the other.

Suddenly it is possible for the two objects to interact. We just need to send one of the objects into the other category. Of course, we have two options. We can either send $c$ into $D$ using $F$ ; or we can send $d$ into $C$ using $G$ .

And now we have two different sets of maps we can consider. We can consider the maps between $c$ and $G (d)$ in $C$ ; or we can consider the maps between $F (c)$ and $d$ in $D$ :

For a random pair of functors $F$ and $G$ , we have no reason to expect any nice relationship between these sets of arrows. As in the visual example above, there might be more arrows on one side of this picture than the other. So let's consider some special cases.

Special Case 1: Isomorphisms of categories

Let's first consider the case in which $F$ and $G$ are mutual inverses. In other words, suppose $G F = I_{C}$ and $F G = I_{D}$ . In this case we say both $F$ and $G$ are isomorphisms and the categories $C$ and $D$ are isomorphic.

Is this case interesting? Not really. An isomorphism of categories means a bijection on both objects and arrows. For all intents and purposes, the categories $C$ and $D$ are identical. Are question about arrows between $c$ and $d$ is then just a question about arrows between two objects in the same category.

Special Case 2: Equivalences of categories

With natural transformations at our disposal, it makes sense to relax the requirement that the compositions $G F$ and $F G$ exactly equal the respective identity functors, and replace it with the weaker condition that the compositions are naturally isomorphic to the respective identity functors. In other words, let's assume there are natural isomorphisms $G F ≃ I_{C}$ and $F G ≃ I_{D}$ . This means that for every $c \in C$ we have an natural isomorphism $c \tilde{\to} G F (c)$ in $C$ , and similarly for every $d \in D$ a natural isomorphism $F G (d) \tilde{\to} d$ in $D$ . In this case, the functors $F$ and $G$ are said to be naturally isomorphic and the categories $C$ and $D$ are said to be equivalent.

When people talk about equivalent categories, this is what they mean. For our purposes, though, this is still too strong of a condition on our functors. Equivalent categories are "basically" the same, and we want to allow for functors between categories that are "very different."

Special Case 3: Adjunction of categories

Suppose we stick to our original idea, and ask for pairs of functors $F, G$ for which arrows $F (c) \to d$ in $D$ are in natural bijection with arrows $c \to G (d)$ in $C$ :

Inverse functors and isomorphic functors certainly satisfy this condition, but potentially many other pairs of functors, too. Following this thread leads to the notion of an adjunction of categories.

Adjunction of categories

There a several equivalent definitions, but we'll give the one that most resembles the situations we've seen:

Definition of an adjunction

Suppose $C$ and $D$ are categories. An adjunction from $C$ to $D$ is a triple $⟨ F, G, τ ⟩ : C ⇀ D$ where $F$ and $G$ are functors in opposite directions between categories $C$ and $D$

$F : C ⇄ D : G$

and $τ$ is a function that assigns to every pair of objects $c \in C$ , $d \in D$ a natural bijection

$τ_{c, d} : {Hom}_{D} (F (c), d) \tilde{\to} {Hom}_{C} (c, G (d)) .$

We call the pair of functors $(F, G)$ and adjoint pair. The functor $F$ is said to be a left adjoint for $G$ , while $G$ is called a right adjoint for $F$ .

We should immediately note that the adjunction requires all three pieces of information: the functor $F$ , the functor $G$ , and the natural bijections in $τ$ between the hom-sets (shown above). In particular, the natural bijections in $τ$ are not implicit in the functors $F$ and $G$ .

The naturality condition

The statement that $τ_{c, d}$ is natural has a formal meaning. Let's first consider "naturality in $c$ ." Suppose $f : c \to c^{'}$ is an arrow in $C$ . The functor $F$ then yields an arrow $F (f) : F (c) \to F (c^{'})$ in $D$ . For each arrow $g : F (c^{'}) \to d$ in $D$ , pre-composition with $F (f)$ yields and arrow $g \circ F (f) : F (c) \to d$ in $D$ . Similarly, for each arrow $h : c^{'} \to G (d))$ , pre-composition with $f$ yields an arrow $h \circ f : c \to G (d)$ in $D$ . Naturality in $c$ is then the requirement that the diagram below commutes:

Similarly, naturality in $d$ is the requirement that for every arrow $g : d \to d^{'}$ in $D$ , the diagram below commutes:

In short, the bijections in $τ$ should be compatible with arrow composition.

Aside

The description of an adjunction can be rephrased in an equivalent way in which $τ$ becomes a bona fide natural isomorphism between two functors.

We will see a plethora of examples (below), but first a few additional notes are in order.

Units and counits

There is a lot of additional information that can be extracted from an adjunction, and some of that information is equivalent to the adjunction itself. Here we list some of that data.

If we set $d = F (c)$ in the natural bijection of the adjunction, then the left-hand hom-set is the set ${Hom}_{D} (F (c), F (c))$ . This set contains one identifiable arrow, namely the identity arrow $1_{F (c)}$ . Let $η_{c}$ denote its image under $τ$ . This arrow is a universal arrow from $c$ to $G$ , and the function $c \mapsto η_{c}$ defines a natural transformation from the identity functor $I_{C}$ to $G F$ :
$η : I_{C} \Rightarrow G F$
We call $η$ the unit of the adjunction. This situation is visualized below:

This unit $η : I_{C} \Rightarrow G F$ is the weakening of the natural isomorphism $G F ≃ I_{C}$ in Case 2, above.

A warning about the unit

The image above might suggest that $η_{c}$ is simply $G (1_{F (c)})$ , but this is not the case. By functoriality we always have $G (1_{F (c)}) = 1_{G F (c)}$ , i.e., it's simply the identity arrow on $G F (c)$ . On the other hand, the unit is an arrow $c \to G F (c)$ , and as we'll see in our examples below, it is often very far from an identity arrow.

Similarly, if we set $c = G (d)$ in the natural bijection above, then the right-hand hom-set contains the identity arrow $1_{G (d)}$ . Let $ε_{d}$ denote its preimage under $τ$ . This arrow is a universal arrow from $F$ to $d$ , and the function $d \mapsto ε_{d}$ defines a natural transformation from $F G$ to the identity functor $I_{D}$ :
$ε : F G \Rightarrow I_{D}$
We call $ε$ the counit of the adjunction. This dual situation is visualized below:

Moreover, the unit and counits satisfy two "triangular" identities, namely the commutative triangles of natural transformations below:

What should you take away from these triangles? Not much. Only that these identities give specific natural isomorphisms of $F ≃ F G F$ and $G ≃ G F G$ . In other words, after each round trip, nothing new really happens. Doing $F$ , then $G$ , then $F$ again is "essentially the same" as just doing $F$ . So while the functors $F$ and $G$ are (probably) not mutual inverses, they're the next closest thing.

In any case, it turns out that an adjunction between $C$ and $D$ can be given equivalently by the data of the two functors $F$ and $G$ , together with the unit $η : I_{C} \to G F$ and the counit $ε F G \to I_{D}$ (satisfying those triangle identities). See page 82 in Mac Lane for details.

Suggested next note

Examples of adjoints

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

Examples of initial, terminal and null objects

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 $C$ 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 $g \mapsto e_{H}$ .

Additive categories

Definition of an additive category

An additive category is an $A b$ -category which has a null object $0$ and a biproduct for each pair of its objects.