Abstract Algebra — Chapter 9 Category theory

Chapter 9
Category theory

9.1. Categories

The extremely broad concept of a “category” allows us to deal with many of the constructions in mathematics in an abstract context. We begin with the definition. We will mostly ignore set-theoretical considerations that can be used to put what follows on a firmer basis, but note that a class is a collection of objects that can be larger than a set, e.g., the class of all sets, in order that we might avoid Russell’s paradox.

Definition 9.1.1. §

A category $𝒞$ is

(1)

a class of objects $Obj (𝒞)$ ,

(2)

for every $A, B \in Obj (𝒞)$ , a class ${Hom}_{𝒞} (A, B)$ of morphisms from $A$ to $B$ , where we often use the notation $f : A \to B$ to indicate that $f$ is an element of ${Hom}_{𝒞} (A, B)$ , and

(3)

a composition map

{Hom}_{𝒞} (A, B) \times {Hom}_{𝒞} (B, C) \to {Hom}_{𝒞} (A, C)

for each $A, B, C \in Obj (𝒞)$ that takes $(f, g)$ for $f : A \to B$ and $g : B \to C$ to the composition $g \circ f$ , subject to the properties that

i.

for each $A \in Obj (𝒞)$ , there exists an identity morphism ${id}_{A} : A \to A$ such that, for all $f : A \to B$ and $g : B \to A$ with $B \in Obj (𝒞)$ , we have

\begin{array}{rcl} f \circ {id}_{A} = f & and & {id}_{A} \circ g = g, \end{array}

and

ii.

composition is associative, i.e.,

h \circ (g \circ f) = (h \circ g) \circ f

for any three morphisms $h : C \to D$ , $g : B \to C$ , and $f : A \to B$ between objects $A, B, C, D \in Obj (𝒞)$ .

Definition 9.1.2. §

We say that a category is small if its class of objects is a set.

Remark 9.1.3. §

What we call a category is often referred to as a locally category, and a category in that terminology allows the morphisms between a pair of objects to form a class, not just a set.

Examples 9.1.4. §

a.: The category $𝐒𝐞𝐭$ which has sets as its objects and maps of sets as its morphisms.
b.: The category $𝐆𝐩$ which has groups as its objects and group homomorphisms as it morphisms.
c.: Similarly, we have categories Ring, the objects of which we take to be the (possibly zero) rings with $1$ and with morphisms the ring homomorphisms that preserve $1$ , and Field.
d.: If $R$ is a ring, then the category $R -mod$ has objects the left $R$ -modules and morphisms the left $R$ -module homomorphisms.
e.: The category Top which has topological spaces as its objects and continuous maps as its morphisms.

We may construct new categories out of old. The following provides a useful example.

Definition 9.1.5. §

Let $𝒞$ and $𝒟$ be categories. The product category $𝒞 \times 𝒟$ is the category with objects the pairs $(C, D)$ with $C \in Obj (𝒞)$ and $D \in Obj (𝒟)$ and morphisms $(f, g) : (C, D) \to (C^{'}, D^{'})$ for any $f : C \to C^{'}$ in $𝒞$ and $g : D \to D^{'}$ in $𝒟$ .

Definition 9.1.6. §

Given a category $𝒞$ , we define the opposite category $𝒞^{op}$ to have the same class of objects as $𝒞$ and

{Hom}_{𝒞^{op}} (A, B) = {Hom}_{𝒞} (B, A)

for $A, B \in Obj (𝒞)$ .

Definition 9.1.7. §

A monoid $G$ is a set with an associative binary operation and an identity element for the operation.

Example 9.1.8. §

Any monoid $G$ gives rise to a category with one object, morphisms equal to the elements of $G$ , and composition law given by multiplication. Then $G^{op}$ is again a monoid with the same elements but the multiplication reversed. A category with one object is also called a monoid, and we have a one-to-one correspondence between monoids and these categories.

We will often have cause to single out a particular class of morphisms in a category known as isomorphisms.

Definition 9.1.9. §

Let $𝒞$ be a category.

a.: A morphism $f : A \to B$ in $𝒞$ is an isomorphism if there exists morphism $g : B \to A$ in $𝒞$ such that $g \circ f = {id}_{A}$ and $f \circ g = {id}_{B}$ .
b.: Two objects $A$ and $B$ in $𝒞$ are said to be isomorphic if there exists an isomorphism $f : A \to B$ in $𝒞$ .
c.: If $f : A \to B$ is a morphism and $g \circ f = {id}_{A}$ (resp., $f \circ g = {id}_{B}$ ), then we say that $g$ is a right inverse (resp., a left inverse) to $f$ . If both $g \circ f = {id}_{A}$ and $f \circ g = {id}_{B}$ , then we say that $g$ is (an) inverse to $f$ , or that $f$ and $g$ are inverse to each other (or mutually inverse, or inverses).

Examples 9.1.10. §

a.: The isomorphisms in $𝐒𝐞𝐭$ are the bijections.
b.: The isomorphisms in $𝐆𝐩$ are the isomorphisms of groups.
c.: The isomorphisms in Top are the homeomorphisms.

Definition 9.1.11. §

a.: A morphism $f : A \to B$ in a category $𝒞$ is a monomorphism if for any $g, h : C \to A$ with $C \in Obj (𝒞)$ , the property that $f \circ g = f \circ h$ implies $g = h$ .
b.: A morphism $f : A \to B$ in a category $𝒞$ is an epimorphism if for any $g, h : B \to C$ with $C \in Obj (𝒞)$ , the property that $g \circ f = h \circ f$ implies $g = h$ .

Examples 9.1.12. §

a.: In $𝐒𝐞𝐭$ and $R -mod$ , a morphism is a monomorphism (resp., epimorphism) if and only if it is injective (resp., surjective).
b.: The natural injection $ℤ \to ℚ$ in Ring is an epimorphism, since a ring homomorphism $ℚ \to R$ is completely determined by its value on $1$ .

Remark 9.1.13. §

A morphism $f : A \to B$ in a category $𝒞$ is a monomorphism if and only if the opposite morphism $f^{op} : B \to A$ in $𝒞^{op}$ is an epimorphism.

We have the following.

Lemma 9.1.14. §

Let $f : A \to B$ and $g : B \to A$ be morphisms in a category $𝒞$ such that $g \circ f = {id}_{A}$ . Then $f$ is a monomorphism and $g$ is an epimorphism.

Proof.

Let $h, k : C \to A$ be morphisms such that $f \circ h = f \circ k$ . Then

k = g \circ f \circ k = g \circ f \circ h = h .

Thus $f$ is a monomorphism. Similarly, $g$ is an epimorphism, or apply Remark 9.1.13. □

In other words, right inverses are monomorphisms and left inverses are epimorphisms.

Definition 9.1.15. §

Let $𝒞$ be a category and $C \in Obj (𝒞)$ .

a.: A subobject of $C$ is a pair $(A, ι)$ consisting of an object $A$ and a monomorphism $ι : A \to C$ .
b.: A quotient of $C$ is a pair $(B, π)$ consisting of an object $B$ and an epimorphism $π : C \to B$ .

Definition 9.1.16. §

A subcategory $𝒞$ of a category $𝒟$ is a category with objects consisting of a subclass of $Obj (𝒟)$ and morphisms ${Hom}_{𝒞} (A, B)$ for $A, B \in Obj (𝒞)$ consisting of a subset of ${Hom}_{𝒟} (A, B)$ containing ${id}_{A}$ for $A = B$ and such that composition maps in $𝒞$ agree with the restriction of the composition maps in $𝒟$ between the same objects.

Examples 9.1.17. §

a.: The category $𝐀𝐛$ of abelian groups with morphisms the group homomorphisms between abelian groups is a subcategory of $𝐆𝐩$ .
b.: The category Field is a subcategory of $𝐑𝐢𝐧𝐠$ .

9.2. Functors

To compare two categories, we need some notion of a map between them. Such maps are referred to as functors. There are two basic types.

Definition 9.2.1. §

Let $𝒞$ and $𝒟$ be categories.

a.

A covariant functor (or simply functor) $F : 𝒞 \to 𝒟$ between two categories $𝒞$ and $𝒟$ is a map of objects $F : Obj (𝒞) \to Obj (𝒟)$ and a map of morphisms

F : {Hom}_{𝒞} (A, B) \to {Hom}_{𝒟} (F (A), F (B))

for each $A, B \in Obj (𝒞)$ such that $F ({id}_{A}) = {id}_{F (A)}$ and $F (g \circ f) = F (g) \circ F (f)$ for all $f : A \to B$ and $g : B \to C$ for each $A, B, C \in Obj (𝒞)$ .

b.

As with a covariant functor, a contravariant functor $F : 𝒞 \to 𝒟$ is again a map on objects, but with maps between sets of morphisms of the form

F : {Hom}_{𝒞} (A, B) \to {Hom}_{𝒟} (F (B), F (A))

that satisfies $F ({id}_{A}) = {id}_{F (A)}$ and $F (g \circ f) = F (f) \circ F (g)$ .

We give some examples of functors.

Examples 9.2.2. §

a.: We have the forgetful functors $𝐆𝐩 \to 𝐒𝐞𝐭$ , $𝐑𝐢𝐧𝐠 \to 𝐒𝐞𝐭$ , and $𝐓𝐨𝐩 \to 𝐒𝐞𝐭$ , which take objects to their underlying sets and morphisms to the corresponding set-theoretic maps.
b.: We have another forgetful functor from $R -mod$ to the category $𝐀𝐛$ of abelian groups.
c.: A homomorphism of monoids $G \to G^{'}$ induces a functor of the corresponding categories, and conversely.
d.: The opposite functor $op : 𝒞 \to 𝒞^{op}$ that is the identity on objects and takes a morphism $f : A \to B$ to its opposite morphism $f^{op} : B \to A$ in $𝒞^{op}$ is contravariant.

Remark 9.2.3. §

A contravariant functor $F : 𝒞 \to 𝒟$ may also be viewed as a covariant functor $𝒞 \to 𝒟^{op}$ , in particular by composing $F$ with the opposite functor $op : 𝒟 \to 𝒟^{op}$ .

Remark 9.2.4. §

A subcategory $𝒞$ of a category $𝒟$ is endowed with a canonical inclusion functor that takes an object of $𝒞$ to the same object of $𝒟$ and is the identity map on morphism.

Definition 9.2.5. §

Let $F : 𝒞 \to 𝒟$ be a functor.

a.: The functor $F$ is called faithful if it is one-to-one on morphisms.
b.: The functor $F$ is called full if it is onto on morphisms.
c.: A functor $F$ is fully faithful if it is both faithful and full.
d.: A subcategory is called a full subcategory if it the corresponding inclusion functor is full.

Remark 9.2.6. §

Every functor takes isomorphisms to isomorphisms.

Remark 9.2.7. §

The inclusion functor attached to a subcategory is always faithful.

Remark 9.2.8. §

A fully faithful functor is sometimes referred to as an embedding of categories, or sometimes a full embedding (and when so, a faithful but not necessarily full functor might instead be referred to as an embedding).

Examples 9.2.9. §

a.: The category $𝐀𝐛$ is a full subcategory of $𝐆𝐩$ .
b.: The category Field is a full subcategory of Ring.
c.: The above-described forgetful functors to sets are faithful but not full.
d.: The category in which the objects are sets but the morphisms are bijections of sets is a subcategory of Set that has the same objects but is not full.

Definition 9.2.10. §

A directed graph $𝒢$ is a collection consisting of

(1): a set $V_{𝒢}$ of vertices of $𝒢$ and,
(2): for every $v, w \in V_{𝒢}$ , a set $E_{𝒢} (v, w)$ of edges from $v$ to $w$ in $𝒢$ .

Terminology 9.2.11. §

In category theory, we often refer to the vertices of a directed graph as dots and the edges as arrows.

Example 9.2.12. §

The following picture provides the data of a directed graph with $4$ vertices and edge sets with between $0$ and $2$ elements each:

A directed graph with four vertices. A full diagram description follows.

Diagram description: A directed graph with four vertices

This is a directed graph. No equality of different paths is asserted.

Objects, listed by row and column:

Row 1, from left to right: column 1: dot; column 2: dot.
Row 2, from left to right: column 1: dot; column 2: dot.

Arrows and lines:

A curved arrow from dot (row 1, column 1) to dot (row 1, column 2), without a label.
An arrow from dot (row 1, column 1) to dot (row 1, column 2), without a label.
An arrow from dot (row 1, column 2) to dot (row 2, column 2), without a label.
An arrow from dot (row 2, column 1) to dot (row 2, column 2), without a label.
A curved arrow from dot (row 2, column 2) to dot (row 1, column 2), without a label.

Definition 9.2.13. §

The category (freely) generated by a directed graph $𝒢$ is the category $I$ with $Obj (I) = V_{𝒢}$ and, for $v, w \in Obj (I)$ , with ${Hom}_{I} (v, w)$ equal to the set of all words $e_{n} e_{n - 1} \dots e_{1}$ for some $n \geq 0$ (with $n = 0$ providing the empty word) with $e_{i} \in E_{𝒢} (v_{i - 1}, v_{i})$ for $v_{i} \in V_{𝒢}$ for $1 \leq i \leq n$ , with $v_{0} = v$ and $v_{n} = w$ , together with the composition given by concatenation of words.

Example 9.2.14. §

Consider the directed graph $𝒢$ given by

A directed path with three vertices. A full diagram description follows.

Diagram description: A directed path with three vertices

This is a directed graph. No equality of different paths is asserted.

Objects, listed by row and column:

Row 1, from left to right: column 1: v subscript (1); column 2: v subscript (2); column 3: v subscript (3).

Arrows and lines:

An arrow from v subscript (1) to v subscript (2), labelled e subscript (1).
An arrow from v subscript (2) to v subscript (3), labelled e subscript (2).

The category $I$ generated by $𝒢$ has three objects $v_{1}, v_{2}, v_{3}$ and morphism sets

\begin{array}{rcl} {Hom}_{I} (v_{i}, v_{i}) = {{id}_{v_{i}}}, & {Hom}_{I} (v_{i}, v_{i + 1}) = {e_{i}}, \\ {Hom}_{I} (v_{1}, v_{3}) = {e_{2} e_{1}}, & and & {Hom}_{I} (v_{i}, v_{j}) = \emptyset if j < i . \end{array}

Example 9.2.15. §

Consider the directed graph $𝒢$ given by

Diagram description: Two vertices with an arrow in each direction

This is a directed graph. No equality of different paths is asserted.

Objects, listed by row and column:

Row 1, from left to right: column 1: v subscript (1); column 2: v subscript (2).

Arrows and lines:

A curved arrow from v subscript (1) to v subscript (2), labelled e subscript (1).
A curved arrow from v subscript (2) to v subscript (1), labelled e subscript (2).

Let $I$ be the category generated $𝒢$ . For $i, j \in {1, 2}$ the set ${Hom}_{I} (v_{i}, v_{j})$ consists of the words with alternating letters $e_{1}$ and $e_{2}$ that start with $e_{i}$ and end with $e_{j}$ (including the empty word if $i = j$ ).

Definition 9.2.16. §

A diagram in $𝒞$ is a functor from a category generated by a graph to $𝒞$ .

Remark 9.2.17. §

Let $𝒢$ be a directed graph, let $I$ be the category generated by $𝒢$ , and let $𝒞$ be a category. Given a map $F : V_{𝒢} \to 𝒞$ and functions $F : E_{𝒢} (v, w) \to {Hom}_{𝒞} (F (v), F (w))$ for each $v, w, \in V_{𝒢}$ , there exists a unique functor $F : I \to 𝒞$ that agrees with $F$ on $V_{𝒢}$ and on $E_{𝒢} (v, w) \subseteq {Hom}_{I} (v, w)$ for every $v, w \in V_{𝒢}$ .

Remark 9.2.18. §

Often, we consider finite graphs, in which every collection of vertices and edges is finite. The resulting diagrams are known as finite diagrams.

Definition 9.2.19. §

A commutative diagram in $𝒞$ is a diagram $F : I \to 𝒞$ , where $I$ is the category generated by a graph, which is a constant function on every set of morphisms.

Example 9.2.20. §

To give a functor from $I$ as in Example 9.2.14 to a category $𝒞$ is to proscribe three objects $A, B, C$ in $𝒞$ and two morphisms $f : A \to B$ and $g : B \to C$ . Thus, such a diagram may be represented by

A \to_{}^{f} B \to_{}^{g} C,

and it is automatically commutative.

Example 9.2.21. §

To give a functor from $I$ as in Example 9.2.15 to a category $𝒞$ is to proscribe two objects $A, B$ in $𝒞$ and two morphisms $f : A \to B$ and $g : B \to A$ . The diagram

Two morphisms in opposite directions. A full diagram description follows.

Diagram description: Two morphisms in opposite directions

The diagram commutes if and only if f composed with g is the identity of B, and g composed with f is the identity of A. These inverse relations are a condition, not an assumption about arbitrary f and g.

Objects, listed by row and column:

Row 1, from left to right: column 1: A; column 2: B.

Arrows and lines:

A curved arrow from A to B, labelled f.
A curved arrow from B to A, labelled g.

is commutative if and only if $f \circ g = {id}_{B}$ and $g \circ f = {id}_{A}$ .

9.3. Natural transformations

Definition 9.3.1. §

Let $F, G : 𝒞 \to 𝒟$ be two (covariant) functors. A natural transformation $η : F ⇝ G$ is a class of morphisms $η_{A} : F (A) \to G (A)$ for each $A \in Obj (𝒞)$ subject to the condition that

Naturality square. A full diagram description follows.

Diagram description: Naturality square

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: F(A); column 2: G(A).
Row 2, from left to right: column 1: F(B); column 2: G(B).

Arrows and lines:

An arrow from F(A) to G(A), labelled eta subscript (A).
An arrow from F(A) to F(B), labelled F(f).
An arrow from G(A) to G(B), labelled G(f).
An arrow from F(B) to G(B), labelled eta subscript (B).

commutes for every $f : A \to B$ and $A, B \in Obj (𝒞)$ . If instead $F$ and $G$ are contravariant functors, then the direction of the vertical arrows in the diagram are reversed.

Example 9.3.2. §

Consider the functor $T : 𝐀𝐛 \to 𝐀𝐛$ that sends an abelian group $A$ to its torsion subgroup $A_{tor}$ (i.e., the subgroup of elements of finite order) and takes a homomorphism $f : A \to B$ to its restriction $T (f) : A_{tor} \to B_{tor}$ . Let $I : 𝐀𝐛 \to 𝐀𝐛$ denote the identity functor. For each abelian group $A$ , we can define $ι_{A} : A_{tor} \to A$ to be the inclusion map. We clearly have $f \circ ι_{A} = ι_{B} \circ T (f)$ for all $f : A \to B$ , so $ι : T ⇝ I$ is a natural transformation.

Example 9.3.3. §

If we think of groups $G$ and $G^{'}$ as monoids, so that functors $G \to G^{'}$ are homomorphisms, then a natural transformation $f ⇝ f^{'}$ between two homomorphisms $f, f^{'} : G \to G^{'}$ is given simply by an element $x \in G^{'}$ such that $f^{'} (g) = 𝑥𝑓 (g) x^{- 1}$ for all $g \in G$ .

Definition 9.3.4. §

Let $F, G : 𝒞 \to 𝒟$ be functors. A natural transformation $η : F ⇝ G$ is said to be a natural isomorphism if each $η_{A}$ for $A \in 𝒞$ is an isomorphism.

Remark 9.3.5. §

Every natural isomorphism $η : F ⇝ G$ has an inverse $η^{- 1} : G ⇝ F$ with $η_{A}^{- 1} = {(η_{A})}^{- 1}$ for $A \in Obj (𝒞)$ .

Definition 9.3.6. §

Let $F, G : 𝒞 \to 𝒟$ be functors, and let $η_{A} : F (A) \to G (A)$ be morphisms for each $A \in Obj (𝒞)$ . We say that these morphisms are natural if the $η_{A}$ form a natural transformation $η : F \to G$ .

Definition 9.3.7. §

Two categories $𝒞$ and $𝒟$ are said to be equivalent if there exist functors $F : 𝒞 \to 𝒟$ and $G : 𝒟 \to 𝒞$ and natural isomorphisms $η : G \circ F ⇝ {id}_{𝒞}$ and $η^{'} : F \circ G ⇝ {id}_{𝒟}$ . Two such functors $F$ and $G$ are said to be quasi-inverse, and $F$ and $G$ are said to be equivalences of categories.

Example 9.3.8. §

A category $𝒞$ with one object $0$ and one morphism is equivalent to the category $𝒟$ with two objects $1$ , $2$ and four morphisms, the identity morphisms of $1$ and $2$ and isomorphisms $1 \to 2$ and $2 \to 1$ . We have quasi-inverse functors $F$ and $G$ with $F (0) = 1$ and $F ({id}_{0}) = {id}_{1}$ and $G (1) = G (2) = 0$ and $G (f) = {id}_{0}$ for all $f$ . To see naturality, note that every morphism between two objects in either category is unique.

The following theorem provides a standard example of equivalence of categories.

Theorem 9.3.9 (Morita equivalence). §

The category of left modules over a ring $R$ is equivalent to the category of left modules over $M_{n} (R)$ for every $n \geq 1$ .

Proof.

Let $A$ be the $R$ - $M_{n} (R)$ -bimodule of row vectors of length $n$ with $R$ -entries. Let $B$ be the $M_{n} (R)$ - $R$ -bimodule of column vectors of length $n$ with $R$ -entries. Define

F : R -mod \to M_{n} (R) -mod, F (M) = B \otimes_{R} M, F (f) = {id}_{B} \otimes f

for left $R$ -modules $M$ and $M^{'}$ and $f \in {Hom}_{R} (M, M^{'})$ . Also, define

G : M_{n} (R) -mod \to R -mod, G (N) = A \otimes_{M_{n} (R)} N, G (g) = {id}_{A} \otimes g

for left $M_{n} (R)$ -modules $N$ and $N^{'}$ and $g \in {Hom}_{M_{n} (R)} (N, N^{'})$ . Since multiplication induces isomorphisms

\begin{array}{rcl} A \otimes_{M_{n} (R)} B \to_{}^{\sim} R & and & B \otimes_{R} A \to_{}^{\sim} M_{n} (R), \end{array}

both $G \circ F$ and $F \circ G$ are naturally isomorphic to identity functors. □

Definition 9.3.10. §

Given two categories $𝒞$ and $𝒟$ with $𝒞$ small, the functor category $𝐅𝐮𝐧𝐜 (𝒞, 𝒟)$ has objects the functors $𝒞 \to 𝒟$ and morphisms the natural transformations between functors, defining composition of natural transformations via composition of the morphisms determining them.

Definition 9.3.11. §

Let $𝒞$ be a category and $A \in Obj (𝒞)$ be an object.

a.

We have a functor $h_{A} : 𝒞 \to Sets$ given by

h_{A} (B) = {Hom}_{𝒞} (A, B)

and, for $g : B \to C$ ,

h_{A} (g) (f) = g \circ f

for all $f : A \to B$ .

b.

We have a contravariant functor $h^{A} : 𝒞 \to 𝐒𝐞𝐭$ with

\begin{array}{rcl} h^{A} (B) = {Hom}_{𝒞} (B, A) & and & h^{A} (g) (f) = f \circ g \end{array}

for $B, C \in Obj (𝒞)$ , $g : B \to C$ , and $f : C \to A$ .

Definition 9.3.12. §

Let $𝒞$ be a small category. The Yoneda embedding is the functor

h^{𝒞} : 𝒞 \to 𝐅𝐮𝐧𝐜 (𝒞^{op}, 𝐒𝐞𝐭)

defined by $h^{𝒞} (A) = h^{A}$ for $A \in Obj (𝒞)$ and $h^{𝒞} (f) : h^{A} ⇝ h^{B}$ for $f : A \to B$ in $𝒞$ given by

h^{𝒞} {(f)}_{C} (g) = f \circ g

for each $g : C \to A$ in $𝒞$ and any $C \in Obj (𝒞)$ .

Remark 9.3.13. §

The reader should check that the Yoneda embedding is a well-defined functor.

Theorem 9.3.14. §

Let $𝒞$ be a small category. The Yoneda embedding $h^{𝒞}$ is fully faithful.

Proof.

We first show faithfulness. Let $f, g : A \to B$ be two morphisms with $h^{𝒞} (f) = h^{𝒞} (g)$ . Then

f = f \circ {id}_{A} = h^{𝒞} {(f)}_{A} ({id}_{A}) = h^{𝒞} {(g)}_{A} ({id}_{A}) = g \circ {id}_{A} = g .

As for fullness, suppose that $η : h^{A} ⇝ h^{B}$ for some $A, B \in Obj (𝒞)$ . We claim that $η = h (e)$ , where $e = η_{A} ({id}_{A})$ . To see this, note that the fact that $η$ is a natural transformation means, in particular, that the diagram

Naturality for represented functors. A full diagram description follows.

Diagram description: Naturality for represented functors

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: h superscript (A)(A); column 2: h superscript (B)(A).
Row 2, from left to right: column 1: h superscript (A)(C); column 2: h superscript (B)(C).

Arrows and lines:

An arrow from h superscript (A)(A) to h superscript (B)(A), labelled eta subscript (A).
An arrow from h superscript (A)(A) to h superscript (A)(C), labelled h superscript (A)(f).
An arrow from h superscript (B)(A) to h superscript (B)(C), labelled h superscript (B)(f).
An arrow from h superscript (A)(C) to h superscript (B)(C), labelled eta subscript (C).

commutes for any $f : C \to A$ . Applying both compositions to the identity morphism of $A$ , we get the two equal terms

h^{B} (f) \circ η_{A} ({id}_{A}) = h^{B} (f) (e) = f \circ e = h {(e)}_{C} (f)

and

η_{C} \circ h^{A} (f) ({id}_{A}) = η_{C} ({id}_{A} \circ f) = η_{C} (f),

and therefore, the desired equality. □

Remark 9.3.15. §

Similarly, we have a fully faithful contravariant functor

h_{𝒞} : 𝒞 \to 𝐅𝐮𝐧𝐜 (𝒞, 𝐒𝐞𝐭)

given by the $h_{A}$ for $A \in Obj (𝒞)$ and natural transformations between them. This is just the Yoneda embedding for the category $𝒞^{op}$ .

Theorem 9.3.14 can be thought of as a more general version of the following standard theorem of group theory.

Corollary 9.3.16 (Cayley’s theorem). §

Every group $G$ is isomorphic to a subgroup of the symmetric group $S_{G}$ on $G$ .

Proof.

Consider the monoid $𝔾$ formed by $G$ . Recall that in $𝔾$ , morphisms are elements of $G$ . As $h : 𝔾 \to 𝐅𝐮𝐧𝐜 (𝔾^{op}, 𝐒𝐞𝐭)$ is a functor, Yoneda’s lemma provides an injective function

h : G \to {Hom}_{𝐅𝐮𝐧𝐜 (𝔾^{op}, 𝐒𝐞𝐭)} (h^{G}, h^{G})

on morphisms with the properties that $h (e) = {id}_{h^{G}}$ and $h (𝑥𝑦) = h (x) \circ h (y)$ for $x, y \in G$ . Since $𝔾$ has only the object $G$ , and $h^{G} (G) = G$ , this induces a one-to-one function $ρ : G \to Maps (G, G)$ with $ρ (x) = h {(x)}_{G}$ and satisfying $ρ (𝑥𝑦) = ρ (x) \circ ρ (y)$ and $ρ (e) = {id}_{G}$ . In particular, we have $ρ (x^{- 1}) \circ ρ (x) = {id}_{G}$ for every $x \in G$ , so its image lands in $S_{G}$ , and the resulting map $G \to S_{G}$ is an injective homomorphism. □

We shall later require the following strengthening of Theorem 9.3.14.

Theorem 9.3.17 (Yoneda’s lemma). §

For any object $A$ of a small category $𝒞$ and contravariant functor $F : 𝒞 \to 𝐒𝐞𝐭$ , there is a bijection

{Hom}_{𝐅𝐮𝐧𝐜 (𝒞^{op}, 𝐒𝐞𝐭)} (h^{A}, F) \to_{}^{\sim} F (A)

given by $η \mapsto η_{A} ({id}_{A})$ that is natural in $A$ and $F$ .

Proof.

Let $B \in Obj (𝒞)$ . Given $x \in F (A)$ , consider the composition

{Hom}_{𝒞} (B, A) \to_{}^{F} {Hom}_{𝐒𝐞𝐭} (F (A), F (B)) \to_{}^{{ev}_{x}} F (A),

where ${ev}_{x}$ is evaluation at $x$ . This defines a natural transformation $ξ^{x} : h^{A} ⇝ F$ . If $η : h^{A} ⇝ F$ and $f : B \to A$ , then

F (f) \circ η_{A} ({id}_{A}) = η_{B} ({id}_{A} \circ f) = η_{B} (f)

by the naturality of $η$ . On the other hand, if $x \in F (A)$ , then

ξ_{A}^{x} ({id}_{A}) = {ev}_{x} (F ({id}_{A})) = {ev}_{x} ({id}_{F (A)}) = x .

Hence the maps $η \mapsto η_{A} ({id}_{A})$ and $x \mapsto ξ^{x}$ are inverse to each other. □

9.4. Limits and colimits

In this section, $𝒞$ denotes a category, and $I$ denotes a small category.

Notation 9.4.1. §

We write $i \in I$ to denote, more simply, that $i$ is an object in $I$ .

Definition 9.4.2. §

Let $F : I \to 𝒞$ be a functor. When it exists, the limit of $F$ is a pair $(\lim F, {(ϕ_{i})}_{i \in I})$ consisting of an object $\lim F$ in $𝒞$ and morphisms

ϕ_{i} : \lim F \to F (i)

for each $i \in I$ such that $ϕ_{j} = F (κ) \circ ϕ_{i}$ for all morphisms $κ : i \to j$ in $I$ and with the universal property that if $X$ is any object of $𝒞$ together with morphisms $ψ_{i} : X \to F (i)$ for which $ψ_{j} = κ \circ ψ_{i}$ for all morphisms $κ : i \to j$ , then there exists a unique morphism $f : X \to \lim F$ such that $ψ_{i} = ϕ_{i} \circ f$ for all $i \in I$ .

Notation 9.4.3. §

We usually use $\lim F$ to refer more simply to a pair $(\lim F, {(ϕ_{i})}_{i \in I})$ that is a limit of $F : I \to 𝒞$ , with the maps understood.

Remark 9.4.4. §

The universal property of the limit of a functor $F$ as in Definition 9.4.2 may be visualized by commutative diagrams

Diagram description: Universal property of a limit

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 2: X.
Row 2, from left to right: column 2: limit F.
Row 3, from left to right: column 1: F(i); column 3: F(j).

Arrows and lines:

An arrow from X to limit F, labelled f.
A curved arrow from X to F(i), labelled psi subscript (i).
A curved arrow from X to F(j), labelled psi subscript (j).
An arrow from limit F to F(i), labelled phi subscript (i).
An arrow from limit F to F(j), labelled phi subscript (j).
An arrow from F(i) to F(j), labelled F( kappa ).

Lemma 9.4.5. §

If $(X, {(ψ_{i})}_{i \in I})$ and $(\lim F, {(ϕ_{i})}_{i \in I})$ are limits of a functor $F : I \to 𝒞$ , then there is a unique isomorphism $f : X \to \lim F$ such that $ψ_{i} = ϕ_{i} \circ f$ for all $i \in I$ .

Proof.

There are morphisms $f : X \to \lim F$ and $g : \lim F \to X$ that are unique with the respective properties that $ψ_{i} = ϕ_{i} \circ f$ and $ϕ_{i} = ψ_{i} \circ g$ for all $i \in I$ . Note that we have $ϕ_{i} = ϕ_{i} \circ f \circ g$ for all $i \in I$ . On the other hand, the universal property of $X$ implies that the identity ${id}_{X}$ is the unique morphism $h$ such that $ψ_{i} \circ h = ψ_{i}$ for all $i \in I$ , so $f \circ g = {id}_{X}$ . Similarly, $g \circ f$ is the identity of $\lim F$ by its universal property. Therefore, the unique map $f$ is an isomorphism. □

Remark 9.4.6. §

Lemma 9.4.5 says that a limit, when it exists, is unique up to unique isomorphism (respecting the universal property) and for that reason, we refer to “the”, rather than “a”, limit.

If $I$ has only identity morphisms, then the limit of a functor $F : I \to 𝒞$ is determined entirely by the image objects $A_{i} = F (i)$ for all $i \in I$ . Hence the notation in the following definition makes sense.

Definition 9.4.7. §

Let $I$ be a category with only identity morphisms, and let $F : I \to 𝒞$ be a functor. Set $A_{i} = F (i)$ for each $i \in I$ .

a.

The limit $\prod_{i \in I} A_{i}$ of $F$ , when it exists, is called the product of the $A_{i}$ .

b.

The maps

p_{i} : \prod_{i \in I} A_{i} \to A_{i}

resulting from the universal property of the product are known as projection maps.

Examples 9.4.8. §

The product coincides with direct product in the categories $𝐒𝐞𝐭$ , $𝐆𝐩$ , Top, Ring and $R -mod$ . Products of more than one object do not exist in the category Field.

Remark 9.4.9. §

A commutative diagram in a category $𝒞$ arises from a functor $F : I \to 𝒞$ , where $I$ is a category generated by a directed graph. Therefore, we may speak of the limit of the diagram.

Definition 9.4.10. §

The limit $A_{1} \times_{B} A_{2}$ of a diagram

The diagram defining a pullback. A full diagram description follows.

Diagram description: The diagram defining a pullback

These objects and maps are the input diagram for the construction described in the surrounding text; no additional equality of its parallel maps is asserted.

Objects, listed by row and column:

Row 1, from left to right: column 2: A subscript (1).
Row 2, from left to right: column 1: A subscript (2); column 2: B.

Arrows and lines:

An arrow from A subscript (1) to B, labelled f subscript (1).
An arrow from A subscript (2) to B, labelled f subscript (2).

(9.4.1)

in $𝒞$ , when it exists, is called the pullback of the diagram.

Remark 9.4.11. §

The pullback of (9.4.1) is endowed with morphisms $p_{1}$ and $p_{2}$ that make

Pullback square. A full diagram description follows.

Diagram description: Pullback square

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: A subscript (1) times subscript (B) A subscript (2); column 2: A subscript (1).
Row 2, from left to right: column 1: A subscript (2); column 2: B.

Arrows and lines:

An arrow from A subscript (1) times subscript (B) A subscript (2) to A subscript (2), labelled p subscript (2).
An arrow from A subscript (1) times subscript (B) A subscript (2) to A subscript (1), labelled p subscript (1).
An arrow from A subscript (1) to B, labelled f subscript (1).
An arrow from A subscript (2) to B, labelled f subscript (2).

commute.

Example 9.4.12. §

In $𝐒𝐞𝐭$ , $𝐆𝐩$ , Top, and $R -mod$ , the pullback is the subobject (i.e., subset, subgroup, subspace, or submodule) with underlying set

{(a_{1}, a_{2}) \in A_{1} \times A_{2} ∣ f_{1} (a_{1}) = f_{2} (a_{2})} .

We also have the dual notion to limits:

Definition 9.4.13. §

Let $F : I \to 𝒞$ be a functor. When it exists, the colimit of $F$ is a pair $(colim F, {(α_{i})}_{i \in I})$ consisting of an object $colim F \in 𝒞$ together with morphisms

α_{i} : F (i) \to colim F

for each $i \in I$ such that $α_{j} \circ F (κ) = α_{i}$ for all morphisms $κ : i \to j$ and with the universal property that if $X$ is any object of $𝒞$ together with morphisms $β_{i} : X \to F (i)$ for which $β_{j} \circ κ = β_{i}$ for all morphisms $κ : i \to j$ , then there exists a unique morphism $f : colim F \to X$ such that $β_{i} = f \circ α_{i}$ for all $i \in I$ .

Notation 9.4.14. §

A colimit of a functor $F : I \to 𝒞$ is usually denoted simply by the object $colim F$ , with the morphisms omitted.

Remark 9.4.15. §

The properties of the colimit expressed in Definition 9.4.13 may be summarized by the commutativity of the diagrams

Universal property of a colimit. A full diagram description follows.

Diagram description: Universal property of a colimit

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: F(i); column 3: F(j).
Row 2, from left to right: column 2: colimit F.
Row 3, from left to right: column 2: X.

Arrows and lines:

A curved arrow from F(i) to X, labelled beta subscript (i).
An arrow from F(i) to colimit F, labelled alpha subscript (i).
An arrow from F(i) to F(j), labelled F( kappa ).
A curved arrow from F(j) to X, labelled beta subscript (j).
An arrow from F(j) to colimit F, labelled alpha subscript (j).
An arrow from colimit F to X, labelled f.

for all $κ : i \to j$ in $I$ .

We have the obvious analogue of Lemma 9.4.5, which again tells us that we may speak of “the” colimit.

Lemma 9.4.16. §

If $(X, {(β_{i})}_{i \in I})$ and $(colim F, {(α_{i})}_{i \in I})$ are colimits of a functor $F : I \to 𝒞$ , then there is a unique isomorphism $f : colim F \to X$ such that $α_{i} = f \circ β_{i}$ for all $i \in I$ .

Remark 9.4.17. §

When it exists, the colimit of $F : I \to 𝒞$ in $𝒞$ satisfies

colim F = op (\lim (op \circ F)),

so its underlying object is an limit in $𝒞^{op}$ .

Definition 9.4.18. §

The colimit of a functor $F : I \to 𝒞$ from a category $I$ with only identity morphisms is called a coproduct, and it is denoted $∐_{i \in I} F (i)$ .

Examples 9.4.19. §

a.: The coproduct in $𝐒𝐞𝐭$ and Top of two objects $X_{1}$ and $X_{2}$ is the disjoint union $X_{1} ∐ X_{2}$ .
b.: The coproduct in $𝐆𝐩$ of two groups $G_{1}$ and $G_{2}$ is the free product $G_{1} * G_{2}$ .
c.: The coproduct in $R -mod$ (and in particular $𝐀𝐛$ ) of two $R$ -modules $A_{1}$ and $A_{2}$ is the direct sum $A_{1} \oplus A_{2}$ .
d.: The coproduct in the category of commutative rings $R_{1}$ and $R_{2}$ is the tensor product $R_{1} \otimes R_{2}$ .

Remark 9.4.20. §

Examples 9.4.19(a-d) generalize directly to arbitrary collections of objects.

Remark 9.4.21. §

Much as with limits, we may speak of a colimit of a diagram in a category.

Definition 9.4.22. §

The colimit of a diagram

The diagram defining a pushout. A full diagram description follows.

Diagram description: The diagram defining a pushout

These objects and maps are the input diagram for the construction described in the surrounding text; no additional equality of its parallel maps is asserted.

Objects, listed by row and column:

Row 1, from left to right: column 1: B; column 2: A subscript (1).
Row 2, from left to right: column 1: A subscript (2).

Arrows and lines:

An arrow from B to A subscript (1), labelled g subscript (1).
An arrow from B to A subscript (2), labelled g subscript (2).

(9.4.2)

in $𝒞$ is called the pushout $A_{1} ∐_{B} A_{2}$ .

Remark 9.4.23. §

The pushout of the diagram (9.4.2) fits into a diagram

Pushout square. A full diagram description follows.

Diagram description: Pushout square

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: B; column 2: A subscript (1).
Row 2, from left to right: column 1: A subscript (2); column 2: A subscript (1) coproduct subscript (B) A subscript (2).

Arrows and lines:

An arrow from B to A subscript (1), labelled g subscript (1).
An arrow from B to A subscript (2), labelled g subscript (2).
An arrow from A subscript (1) to A subscript (1) coproduct subscript (B) A subscript (2), labelled iota subscript (1).
An arrow from A subscript (2) to A subscript (1) coproduct subscript (B) A subscript (2), labelled iota subscript (2).

where $ι_{1}$ and $ι_{2}$ are induced by the universal property of the colimit.

Example 9.4.24. §

In $𝐒𝐞𝐭$ and Top, the pushout is the quotient (set or topological space) of the disjoint union of $A_{1}$ and $A_{2}$ under the equivalence relation identifying $g_{1} (b)$ with $g_{2} (b)$ for all $b \in B$ .

Definition 9.4.25. §

We say that a category $𝒞$ admits the limit (resp., colimit) of a functor $F : I \to 𝒞$ if the limit (resp., colimit) exists in $𝒞$ .

Remark 9.4.26. §

More generally, we may speak of $𝒞$ admitting the limits (or colimits) of any collection of functors from small categories to $𝒞$ .

Definition 9.4.27. §

A category is called complete if it admits all limits.

Example 9.4.28. §

The category of finite sets is not complete.

Definition 9.4.29. §

A category is called cocomplete if it admits all colimits.

Remark 9.4.30. §

To say that $𝒞$ is complete is to say that $𝒞^{op}$ is cocomplete.

Proposition 9.4.31. §

The category $𝐒𝐞𝐭$ is both complete and cocomplete.

Proof.

Let $F : I \to 𝐒𝐞𝐭$ be a functor. We merely describe the limit and colimit of $F$ and leave the rest to the reader. The limit is

\lim F = {{(a_{i})}_{i} \in \prod_{i \in I} F (i) ∣ F (ϕ) (a_{i}) = a_{j} if ϕ : i \to j in I},

and the colimit is

colim F = ∐_{i \in I} F (i) ∕ \sim

where $\sim$ is the minimal equivalence relation satisfying $a_{i} \sim a_{j}$ for $a_{i} \in F (i)$ and $a_{j} \in F (j)$ if there exists $ϕ : i \to j$ with $F (ϕ) (a_{i}) = a_{j}$ . □

Remark 9.4.32. §

In fact, the categories $𝐒𝐞𝐭$ , $𝐓𝐨𝐩$ , $𝐆𝐩$ , $𝐀𝐛$ , $𝐑𝐢𝐧𝐠$ , and $R -mod$ admit all limits and colimits.

The reader will easily verify the following.

Proposition 9.4.33. §

Let $I$ be a small category and $𝒞$ a (co)complete category. Then the category $𝐅𝐮𝐧𝐜 (I, 𝒞)$ is (co)complete.

Corollary 9.4.34. §

Let $I$ be a small category and $𝒞$ be (co)complete. Let $F : I \to 𝒞$ be a functor, and supposing that $𝒞$ is small, consider the Yoneda embedding $h^{𝒞} : 𝒞 \to 𝐅𝐮𝐧𝐜 (𝒞^{op}, 𝐒𝐞𝐭)$ . Then $h^{𝒞} \circ F$ has a (co)limit in $𝐅𝐮𝐧𝐜 (𝒞^{op}, 𝐒𝐞𝐭)$ .

Definition 9.4.35. §

A directed set $I$ is a set $I$ together with a partial ordering $\leq$ on $I$ such that for any $i, j \in I$ , there exists $k \in I$ with $i \leq k$ and $j \leq k$ .

Definition 9.4.36. §

a.

The limit of a diagram

\dots \to A_{3} \to A_{2} \to A_{1}

in a category $𝒞$ is referred to as the sequential limit of the objects $A_{i}$ .

b.

The colimit of a diagram

A_{1} \to A_{2} \to A_{3} \to \dots

in a category $𝒞$ is referred to as the sequential colimit of the objects $A_{i}$ .

Example 9.4.37. §

In $𝐀𝐛$ , the sequential limit of the groups $ℤ ∕ p^{n} ℤ$ with respect to homomorphisms $ℤ ∕ p^{n + 1} ℤ \to ℤ ∕ p^{n} ℤ$ given by reduction modulo $p^{n}$ is the group $ℤ_{p}$ of $p$ -adic integers. The sequential colimit of these same groups with respect to the maps $ℤ ∕ p^{n} ℤ \to ℤ ∕ p^{n + 1} ℤ$ induced by multiplication modulo $p$ is the group $ℚ_{p} ∕ ℤ_{p}$ , equal to the $p$ -power torsion in $ℚ ∕ ℤ$ .

The sequential limit (resp., sequential colimit) is just a special case of the notion of an inverse limit (resp., direct limit), which is a more usual terminology.

Definition 9.4.38. §

a.: A directed category $I$ is a category with a nonempty directed set $I$ of objects and at most one morphism $i \to j$ for any $i, j \in I$ , which exists if and only if $i \leq j$ .
b.: A codirected category is a category $I$ such that $I^{op}$ is directed.

Definition 9.4.39. §

Let $I$ be a codirected category. The limit of a functor $F : I \to 𝒞$ is referred to the inverse limit of the objects $F (i)$ over the inverse system of objects $F (i)$ for $i \in I$ and morphisms $F (κ)$ for $κ : i \to j$ in $I$ , and it is denoted ${lim_{\leftarrow}}_{i \in I} F (i)$ .

Definition 9.4.40. §

Let $I$ be a directed category. The colimit of a functor $F : I \to 𝒞$ is the direct limit of the objects $F (i)$ over the directed system of objects $F (i)$ for $i \in I$ and morphisms $F (κ)$ for $κ : i \to j$ in $I$ and is denoted ${lim_{\to}}_{i \in I} F (i)$ (or sometimes just ${lim_{\to}}_{} F$ ).

Examples 9.4.41. §

a.: The inverse limit of the (commutative) rings $ℤ ∕ p^{n} ℤ$ with respect to homomorphisms $ℤ ∕ p^{n + 1} ℤ \to ℤ ∕ p^{n} ℤ$ given by reduction modulo $p^{n}$ is the ring $ℤ_{p}$ known as the $p$ -adic integers.
b.: The direct limit of the abelian groups $ℤ ∕ p^{n} ℤ$ with respect to the multiplication-by- $p$ maps $ℤ ∕ p^{n} ℤ \to ℤ ∕ p^{n + 1} ℤ$ is equal to the subgroup of elements of $p$ -power order (under addition) in $ℚ ∕ ℤ$ .
c.: The absolute group $Gal (\bar{ℚ} ∕ ℚ)$ of all automorphisms of the field of algebraic numbers $\bar{ℚ}$ is isomorphic to the inverse limit of the collection of all $Gal (K ∕ ℚ)$ for $K$ a finite, normal extension of $ℚ$ inside $\bar{ℚ}$ , with respect to the maps $Gal (L ∕ ℚ) \to Gal (K ∕ ℚ)$ given by restriction with $K \subseteq L$ . Note that the set of field extensions is directed as the compositum of two normal extensions is normal, and the resulting category is codirected as we use only the restriction morphisms.

For certain diagrams, limits and colimits can be quite boring, especially when the diagram contains an initial object in the case of limits, or terminal object in the case of colimits.

Definition 9.4.42. §

a.: An initial object $A$ in $𝒞$ is an object such that for each $B \in Obj (𝒞)$ , there is a unique morphism $A \to B$ in $𝒞$ .
b.: A terminal object $X$ in $𝒞$ is an object such that for each $B \in Obj (𝒞)$ , there is a unique morphism $B \to X$ in $𝒞$ .
c.: An zero object $0$ in $𝒞$ is an object that is both initial and terminal.

Remark 9.4.43. §

Terminal and initial objects are unique up to unique isomorphism when defined.

We provide some examples.

Examples 9.4.44. §

a.: The empty set $∅$ is the initial object in the category $𝐒𝐞𝐭$ , while any set with one element is a terminal object.
b.: The trivial group is a zero object in the category $𝐆𝐩$ .
c.: The zero ring is a terminal object and $ℤ$ is an initial object in Ring.

We omit the proof of the following easy lemma.

Lemma 9.4.45. §

Let $I$ be a small category and $F : I \to 𝒞$ a functor.

a.: Suppose that $I$ has an initial object $i$ . Then
$\lim F = F (i) .$
b.: Suppose that $I$ has a terminal object $j$ . Then
$colim F = F (j) .$

9.5. Adjoint functors

The following definition allows us to weaken the property of being quasi-inverse to one of “adjointness”.

Definition 9.5.1. §

We say that $F : 𝒞 \to 𝒟$ is left adjoint to $G : 𝒟 \to 𝒞$ if there exist bijections

ϕ_{C, D} : {Hom}_{𝒟} (F (C), D) \to_{}^{\sim} {Hom}_{𝒞} (C, G (D))

for each $C \in Obj (𝒞)$ and $D \in Obj (𝒟)$ such that the $ϕ_{C, D}$ form a natural transformation of functors

𝒞^{op} \times 𝒟 \to 𝐒𝐞𝐭 .

We also say that $G$ is right adjoint to $F$ , and we say that $F$ and $G$ are adjoint functors.

Remark 9.5.2. §

To say that $η$ is a natural transformation in Definition 9.5.1 is a fancier way of saying that given morphisms $f : C^{'} \to C$ in $𝒞$ and $g : D \to D^{'}$ in $𝒟$ , we have a commutative diagram

Naturality of an adjunction. A full diagram description follows.

Diagram description: Naturality of an adjunction

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: Hom subscript (script D)(F(C),D); column 2: Hom subscript (script C)(C,G(D)).
Row 2, from left to right: column 1: Hom subscript (script D)(F(C prime ),D prime ); column 2: Hom subscript (script C)(C prime ,G(D prime )).

Arrows and lines:

An arrow from Hom subscript (script D)(F(C),D) to Hom subscript (script C)(C,G(D)), labelled eta subscript ((C,D)).
An arrow from Hom subscript (script D)(F(C),D) to Hom subscript (script D)(F(C prime ),D prime ), labelled t maps to g composed with t composed with F(f).
An arrow from Hom subscript (script C)(C,G(D)) to Hom subscript (script C)(C prime ,G(D prime )), labelled u maps to G(g) composed with u composed with f.
An arrow from Hom subscript (script D)(F(C prime ),D prime ) to Hom subscript (script C)(C prime ,G(D prime )), labelled eta subscript ((C prime ,D prime )).

Remark 9.5.3. §

If $F : 𝒞 \to 𝒢$ and $G : 𝒟 \to 𝒞$ are quasi-inverse functors, then we have bijections

ϕ_{C, D} : {Hom}_{𝒟} (F (C), D) \to_{}^{\sim} {Hom}_{𝒞} (C, G (D)), ϕ_{C, D} (g) = G (g) \circ η_{C},

for $C \in Obj (𝒞)$ and $D \in Obj (𝒟)$ , where $g : F (C) \to D$ in $𝒟$ and $η$ is a natural isomorphism $G \circ F ⇝ {id}_{𝒞}$ . These form a natural transformation $ϕ$ between functors $𝒞^{op} \times 𝒟 \to 𝐒𝐞𝐭$ , so $G$ is right adjoint to $F$ . Similarly, using $η^{'} : F \circ G ⇝ {id}_{𝒟}$ , we see that $G$ is left adjoint to $F$ .

Example 9.5.4. §

The forgetful functor $𝐆𝐩 \to 𝐒𝐞𝐭$ is right adjoint to the functor $𝐒𝐞𝐭 \to 𝐆𝐩$ that takes a set $X$ to the free group $F_{X}$ on $X$ and a map $f : X \to Y$ of sets to the unique homomorphism $ϕ_{f} : F_{X} \to F_{Y}$ with $ϕ_{f} |_{X} = f$ . That is, the restriction map

Hom (F_{Y}, G) \to Maps (Y, G)

is inverse to the map $Maps (Y, G) \to Hom (F_{Y}, G)$ that takes $ϕ_{f} : Y \to G$ to $F (f)$ , and these bijections are easily seen to be natural.

The following is perhaps the most standard example of adjointness: that of $Hom$ and $\otimes$ in categories of modules.

Example 9.5.5. §

Let $A$ and $B$ be algebras over a commutative ring $R$ . Fix an $R$ -balanced $A$ - $B$ -bimodule $M$ . Define a functor $t_{M} : B -mod \to A -mod$ by $t_{M} (N) = M \otimes_{B} N$ on $B$ -modules $N$ and $t_{M} (g) = {id}_{M} \otimes g$ on $B$ -module homomorphisms $g : N \to N^{'}$ . Define another functor $h_{M} : A -mod \to B -mod$ by $h_{M} (L) = {Hom}_{A} (M, L)$ on $A$ -modules $L$ and $h_{M} (f) (h) = f \circ h$ for $f \in {Hom}_{A} (L, L^{'})$ and $h \in {Hom}_{A} (M, L)$ . Then the isomorphism of Theorem 8.4.29 is the adjunction map

{Hom}_{A} (t_{M} (N), L) \to_{}^{\sim} {Hom}_{B} (N, h_{M} (L)) .

These isomorphisms are natural in $N$ and $L$ , and hence $t_{M}$ is left adjoint to $h_{M}$ .

Proposition 9.5.6. §

Fix categories $I$ and $𝒞$ , and suppose that all limits $F : I \to 𝒞$ exist. The functor $\lim$ has a left adjoint $Δ$ given by taking $A \in Obj (𝒞)$ to the constant functor $c_{A}$ , where $c_{A} (i) = A$ for all $i \in I$ , and taking $g : A \to B$ for $A, B \in Obj (𝒞)$ to the natural transformation $c_{A} ⇝ c_{B}$ given by $g : c_{A} (i) = A \to c_{B} (i) = B$ for all $i \in I$ .

Proof.

We must describe natural isomorphisms

{Hom}_{𝐅𝐮𝐧𝐜 (I, 𝒞)} (c_{A}, F) ≅ {Hom}_{𝒞} (A, \lim F)

for $A \in Obj (𝒞)$ and $F : I \to 𝒞$ . I.e., given a natural tranformation $η : c_{A} ⇝ F$ , we must associate a map $f : A \to \lim F$ , and conversely. Such a natural transformation $η$ consists of maps

η_{i} : c_{A} (i) = A \to F (i)

that are compatible in the sense that $η_{j} = F (κ) \circ η_{i}$ for all $κ : i \to j$ . Thus, the existence of a unique $f$ is simply the universal property of the limit. On the other hand, if we have $f$ , then we have maps

ϕ_{i} \circ f : A \to F (i),

where $ϕ_{i}$ is the map $\lim F \to F (i)$ arising in the definition of the limit. These maps then define the universal transformation $η$ . □

We now see exactly how adjointness weakens inverseness.

Definition 9.5.7. §

Example 9.5.8. §

Notation 9.5.9. §

Let $η : F ⇝ F^{'}$ be a natural transformation between functors $F, F^{'} : 𝒞 \to 𝒟$ .

a.

If $G : 𝒟 \to E$ is a functor, then we define a natural transformation $G (η) : G \circ F ⇝ G \circ F^{'}$ by

G {(η)}_{C} = G (η_{C}) : G (F (C)) \to G (F^{'} (C))

for all objects $C$ of $𝒞$ .

b.

If $H : B \to 𝒞$ is a functor, then we define a natural transformation $η (H) : F \circ H ⇝ F^{'} \circ H$ by

η {(H)}_{B} = η_{H (B)} : F (H (B)) \to F^{'} (H (B))

for all objects $B$ of $𝒞$ .

Definition 9.5.10. §

Let $F : 𝒞 \to 𝒟$ and $G : 𝒟 \to 𝒞$ be functors.

a.

A unit for the pair $(F, G)$ is a natural transformation ${id}_{𝒞} ⇝ G \circ F$ .

b.

A counit for the pair $(F, G)$ is a natural transformation $F \circ G ⇝ {id}_{𝒟}$ .

c.

A unit-counit adjunction is a pair $(F, G)$ , a unit $η$ for $(F, G)$ , and a counit $η^{'}$ for $(F, G)$ satisfying

{id}_{F} = η^{'} (F) \circ F (η) : F ⇝ F

as morphisms in $𝐅𝐮𝐧𝐜 (𝒞, 𝒟)$ and

{id}_{G} = G (η^{'}) \circ η (G) : G ⇝ G

as morphisms in $𝐅𝐮𝐧𝐜 (𝒟, 𝒞)$ .

Proposition 9.5.11. §

A functor $F : 𝒞 \to 𝒟$ is left adjoint to a functor $G : 𝒟 \to 𝒞$ if and only if there exists a unit-counit adjunction for the pair $(F, G)$ .

Proof.

Suppose that $F$ is left adjoint to $G$ . We define $η : {id}_{𝒞} ⇝ G \circ F$ as follows. For $C \in Obj (𝒞)$ , we have bijections

{Hom}_{𝒟} (F (C), F (C)) \to_{}^{\sim} {Hom}_{𝒞} (C, G \circ F (C))

by adjointness, and we define $η_{C}$ to be the image of ${id}_{F (C)}$ . For $D \in Obj (𝒟)$ , we also have

{Hom}_{𝒟} (F \circ G (D), D) \to_{}^{\sim} {Hom}_{𝒞} (G (D), G (D))

and define $η^{'} : F \circ G ⇝ {id}_{𝒟}$ by taking $η_{D}^{'}$ to be the image of ${id}_{G (D)}$ under the inverse of this map. We leave it to the reader to check that these are natural and form a unit-counit adjunction. The converse is left to the reader as well. □

9.6. Representable functors

Definition 9.6.1. §

Let $F : 𝒞 \to 𝐒𝐞𝐭$ be a contravariant functor. Then $F$ is said to be representable if there exists a natural isomorphism $h^{B} ⇝ F$ for some $B \in Obj (𝒞)$ . (In other words, we have natural bijections

{Hom}_{𝒞} (A, B) \to_{}^{\sim} F (A)

for all objects $A$ of $𝒞$ .) We then say that $B$ represents $F$ .

Using Yoneda’s lemma and assuming $𝒞$ to be small, we can reword Definition 9.6.1 as saying that there exists $B \in Obj (𝒞)$ such that there are compatible bijections between the set of natural transformations $h^{A} ⇝ F$ and the set of morphisms $A \to B$ for each $A \in Obj (𝒞)$ .

Example 9.6.2. §

Consider the contravariant functor $P : 𝐒𝐞𝐭 \to 𝐒𝐞𝐭$ which takes a set $S$ to its power set $P (S)$ , the set of all subsets of $S$ and a map $f : S \to T$ to the map $P (f) : P (T) \to P (S)$ by mapping $U \subset T$ to $f^{- 1} (U)$ . Then $P$ is represented by the set ${0, 1}$ via the isomorphism

Maps (S, {0, 1}) \to_{}^{\sim} P (S)

by $ϕ \mapsto ϕ^{- 1} ({1})$ . These isomorphisms form a natural transformation:

Representation of the power-set functor. A full diagram description follows.

Diagram description: Representation of the power-set functor

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: Maps (T,left brace0,1right brace); column 2: P(T).
Row 2, from left to right: column 1: Maps (S,left brace0,1right brace); column 2: P(S).

Arrows and lines:

An arrow from Maps (T,left brace0,1right brace) to P(T), labelled isomorphism symbol.
An arrow from Maps (T,left brace0,1right brace) to Maps (S,left brace0,1right brace), labelled h superscript (left brace0,1right brace)(f).
An arrow from P(T) to P(S), labelled P(f).
An arrow from Maps (S,left brace0,1right brace) to P(S), labelled isomorphism symbol.

for $f : S \to T$ . Here, the lefthand vertical map takes $ϕ$ to $ϕ \circ f$ and the righthand vertical map takes a subset $X$ of $T$ to $f^{- 1} (X)$ . We check that

{(ϕ \circ f)}^{- 1} ({1}) = f^{- 1} (ϕ^{- 1} ({1})) .

The following is a corollary of Yoneda’s lemma.

Lemma 9.6.3. §

A representable functor is represented by a unique object up to isomorphism. If $B$ and $C$ represent a contravariant functor $F : 𝒞 \to 𝐒𝐞𝐭$ , then such an isomorphism $f : B \to C$ is unique making the diagrams

Uniqueness of a representing object. A full diagram description follows.

Diagram description: Uniqueness of a representing object

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: Hom subscript (script C)(A,B); column 2: F(A).
Row 2, from left to right: column 1: Hom subscript (script C)(A,C); column 2: F(A).

Arrows and lines:

An arrow from Hom subscript (script C)(A,B) to F(A) (row 1, column 2), labelled isomorphism symbol.
An arrow from Hom subscript (script C)(A,B) to Hom subscript (script C)(A,C), labelled h superscript (A)(f).
Equality joins F(A) (row 1, column 2) and F(A) (row 2, column 2), without a label.
An arrow from Hom subscript (script C)(A,C) to F(A) (row 2, column 2), labelled isomorphism symbol.

commute for all $A \in Obj (𝒞)$ .

Proof.

Let $F : 𝒞 \to 𝐒𝐞𝐭$ be a representable (contravariant) functor represented by $B \in Obj (𝒞)$ and $C \in Obj (𝒞)$ . Then we have natural isomorphisms $ξ : h^{B} ⇝ F$ and $ξ^{'} : h^{C} ⇝ F$ . The composition $ξ^{'} \circ ξ^{- 1} : h^{B} \to h^{C}$ is equal to $h^{𝒞} (f)$ for a unique $f : B \to C$ by the weak form of Yoneda’s lemma. □

Theorem 9.6.4. §

Let $F : 𝒞 \to 𝒟$ be a functor between small categories.

a.

The functor $F$ has a right adjoint if and only if the functor $h^{D} \circ F$ is representable for each $D \in Obj (𝒟)$ . If $G$ is right adjoint to $F$ , then $h^{D} \circ F$ is representable by $G (D)$ .

b.

If $F$ has a right adjoints $G$ and $G^{'}$ , then there exists a unique natural isomorphism $ξ : G ⇝ G^{'}$ such that diagrams

Uniqueness of a right adjoint. A full diagram description follows.

Diagram description: Uniqueness of a right adjoint

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: Hom subscript (script D)(F(C),D); column 2: Hom subscript (script C)(C,G(D)).
Row 2, from left to right: column 1: Hom subscript (script D)(F(C),D); column 2: Hom subscript (script C)(C,G prime (D)).

Arrows and lines:

An arrow from Hom subscript (script D)(F(C),D) (row 1, column 1) to Hom subscript (script C)(C,G(D)), labelled eta subscript ((C,D)).
Equality joins Hom subscript (script D)(F(C),D) (row 1, column 1) and Hom subscript (script D)(F(C),D) (row 2, column 1), without a label.
An arrow from Hom subscript (script C)(C,G(D)) to Hom subscript (script C)(C,G prime (D)), labelled f maps to xi subscript (D) composed with f.
An arrow from Hom subscript (script D)(F(C),D) (row 2, column 1) to Hom subscript (script C)(C,G prime (D)), labelled eta prime subscript ((C,D)).

commute for all $C \in Obj (𝒞)$ and $D \in Obj (𝒟)$ , where the horizontal morphisms are the adjunction isomorphisms.

Proof.

Assume that $F$ has a right adjoint $G$ , and consider the adjunction morphisms

η_{(C, D)} : {Hom}_{𝒟} (F (C), D) \to_{}^{\sim} {Hom}_{𝒞} (C, G (D)) .

In other words,

h^{D} \circ F (C) ≅ h^{G (D)} (C),

so $G (D)$ represents $h^{D} \circ F$ . In this case, the uniqueness in part b is an immediate consequence of Lemma 9.6.3.

Now suppose that $h^{D} \circ F$ is representable for each $D$ by some object $G (D)$ (chosen using the axiom of choice). Then there exist isomorphisms $η_{(C, D)}$ that are natural in $C$ . We must also define $G$ on morphisms $f : D \to D^{'}$ in $𝒟$ . Such an $f$ induces a natural transformation $h^{D} ⇝ h^{D^{'}}$ which provides morphisms

h^{D} \circ F (C) \to h^{D^{'}} \circ F (C)

for all $C \in Obj (𝒞)$ and thus induces $h^{G (D)} (C) \to h^{G (D^{'})} (C)$ , and these are natural in $C$ . Thus, we have a natural transformation $h^{G (D)} ⇝ h^{G (D^{'})}$ . Since the Yoneda embedding is fully faithful, we have a unique morphism $G (D) \to G (D^{'})$ inducing this natural transformation, which we define to be $G (f)$ . We leave to the reader the check that $G$ as defined is a functor. □

Definition 9.6.5. §

Let $F : 𝒞 \to 𝐒𝐞𝐭$ be a covariant functor. We say that $F$ is representable if there exists a natural isomorphism $h_{A} ⇝ F$ for some $A \in Obj (𝒞)$ . (That is, there are natural isomorphisms

F (B) \to_{}^{\sim} {Hom}_{𝒞} (A, B)

in $B \in Obj (𝒞)$ .) In this case, we say that $A$ represents $F$ .

Remark 9.6.6. §

A covariant functor $F : 𝒞 \to 𝐒𝐞𝐭$ is representable if and only if the contravariant functor $F \circ op : 𝒞^{op} \to 𝐒𝐞𝐭$ is representable. The same object of $𝒞$ will represent both objects.

Example 9.6.7. §

Let $F : 𝐆𝐩 \to 𝐒𝐞𝐭$ be the forgetful functor. Then $F$ can be represented by $ℤ$ . To see this, we define the set map

G \to {Hom}_{𝐆𝐩} (ℤ, G)

by $a \mapsto (1 \mapsto a)$ for $a \in G$ . Naturality is clear.

Example 9.6.8. §

Let $F : 𝐆𝐩 \to 𝐒𝐞𝐭$ be the functor which sends a group to its subset $G [n]$ of elements of order dividing $n$ . Then $F$ can be represented by $ℤ ∕ 𝑛ℤ$ .

Example 9.6.9. §

Consider a functor $F : I \to 𝒞$ between small categories. To say that the contravariant functor $\lim h^{𝒞} \circ F : 𝒞 \to 𝐒𝐞𝐭$ is representable is exactly to say that there exists an object $X$ in $𝒞$ such that one has natural isomorphisms

{Hom}_{𝒞} (A, X) \to_{}^{\sim} \lim (h^{𝒞} \circ F) (A) \to_{}^{\sim} \lim ({Hom}_{𝒞} (A, F (\cdot)))

for $A \in 𝒞$ . In other words, $\lim h^{𝒞} \circ F$ is representable if and only if $\lim F$ exists in $𝒞$ .

Example 9.6.10. §

Consider a functor $F : I \to 𝒞$ between small categories. View $h_{𝒞}$ as a covariant functor $𝒞 \to Hom (𝒞, 𝐒𝐞𝐭)$ . To say that the functor $\lim (h_{𝒞} \circ F) : 𝒞 \to 𝐒𝐞𝐭$ is representable is exactly to say that there exists an object $X \in 𝒞$ such that one has natural isomorphisms

{Hom}_{𝒞} (X, A) \to_{}^{\sim} \lim (h_{𝒞} \circ F) (A) \to_{}^{\sim} \lim ({Hom}_{𝒞} (F (\cdot), A))

for $A \in 𝒞$ . In other words, $\lim h_{𝒞} \circ F$ is representable if and only if $colim F$ exists in $𝒞$ .

9.7. Equalizers and images

Definition 9.7.1. §

Let $𝒞$ be a category, and let

Two parallel morphisms. A full diagram description follows.

Diagram description: Two parallel morphisms

These objects and maps are the input diagram for the construction described in the surrounding text; no additional equality of its parallel maps is asserted.

Objects, listed by row and column:

Row 1, from left to right: column 1: A; column 2: B.

Arrows and lines:

An arrow from A to B, labelled f.
An arrow from A to B, labelled g.

(9.7.1)

be a diagram in $𝒞$ .

a.: The limit $eq (f, g)$ of the diagram (9.7.1), when it exists, is called its equalizer.
b.: The colimit $coeq (f, g)$ of (9.7.1) is called its coequalizer.

We have a commutative diagram:

Equalizer and coequalizer maps. A full diagram description follows.

Diagram description: Equalizer and coequalizer maps

The first arrow is the equalizer arrow: precomposing f and g with it gives equal maps. The last arrow is the coequalizer arrow: postcomposing f and g with it gives equal maps. The parallel arrows f and g need not themselves be equal.

Objects, listed by row and column:

Row 1, from left to right: column 1: equalizer (f,g); column 2: A; column 3: B; column 4: coequalizer (f,g).

Arrows and lines:

An arrow from equalizer (f,g) to A, without a label.
An arrow from A to B, labelled f.
An arrow from A to B, labelled g.
An arrow from B to coequalizer (f,g), without a label.

Examples 9.7.2. §

a.

Let $X, Y$ be sets, and consider maps $f, g : X \to Y$ . In $𝐒𝐞𝐭$ , we have

eq (f, g) = {x \in X ∣ f (x) = g (x)}

and $coeq (f, g)$ is the quotient of $Y$ by the minimal equivalence relation $\sim$ generated by $f (x) \sim g (x)$ for all $x \in X$ .

b.

In $R -mod$ , the equalizer is expressed as in $𝐒𝐞𝐭$ . For an $R$ -module homomorphism $f : A \to B$ , we have

coeq (f, g) = B ∕ {(f - g) (a) ∣ a \in A} .

Lemma 9.7.3. §

Let $f, g : A \to B$ be morphisms in a category $𝒞$ .

a.: Suppose that $eq (f, g)$ exists. Then the induced map $h : eq (f, g) \to A$ is a monomorphism.
b.: Suppose that $coeq (f, g)$ exists. Then the induced map $k : B \to coeq (f, g)$ is an epimorphism.

Proof.

Suppose that $α, β : C \to eq (f, g)$ are morphisms in $𝒞$ such that $h \circ α = h \circ β$ . Let $h^{'} = h \circ α$ , and note that $f \circ h^{'} = g \circ h^{'}$ . But then $α : C \to eq (f, g)$ is unique such that $h^{'} = h \circ α$ by the universal property of $eq (f, g)$ . Since $h^{'} = h \circ β$ as well, we have $α = β$ . Part b follows from part a by working in the opposite category. □

Theorem 9.7.4. §

A category that admits all products and equalizers is complete, and a category that admits all coproducts and coequalizers is cocomplete.

Proof.

For the second statement, by taking the opposite category, we are reduced to the first statement. Let $𝒞$ be a category that admits all products and equalizers. Let $F : I \to 𝒞$ be a functor from a small category $I$ , and consider the equalizer $eq (f, g)$ of the two morphisms

f, g : \prod_{i \in I} F (i) \to \prod_{ϕ : i \to ϕ (i)} F (ϕ (i))

defined via the universal property of the second product as the unique morphisms satisfying

p_{ϕ} \circ f = p_{ϕ (i)} : \prod_{j \in I} F (j) \to F (ϕ (i))

and

p_{ϕ} \circ g = F (ϕ) \circ p_{i} : \prod_{j \in I} F (j) \to F (ϕ (i))

for morphisms $ϕ : i \to ϕ (i)$ in $𝒞$ , where $p_{ϕ}$ denotes projection onto the $ϕ$ -coordinate in the second product and $p_{i}$ denotes projection to the $i$ -coordinate in the first.

Let $ι : eq (f, g) \to \prod_{i \in I} F (i)$ be the morphism defining the equalizer. We claim that the equalizer $eq (f, g)$ , together with the maps $p_{i} \circ ι : eq (f, g) \to F (i)$ for $i \in I$ , satisfies the univeral property of $\lim F$ . By definition, for any morphism $ϕ$ in $𝒞$ as above, we have

F (ϕ) \circ (p_{i} \circ ι) = p_{ϕ} \circ g \circ ι = p_{ϕ} \circ f \circ ι = p_{ϕ (i)} \circ ι .

Moreover, given $X \in 𝒞$ and morphisms $ψ_{i} : X \to F (i)$ for $i \in I$ such that $F (ϕ) \circ ψ_{i} = ϕ_{ϕ (i)}$ for all $i \in I$ , we have a product map $ψ : X \to \prod_{i \in I} F (i)$ such that

p_{ϕ} \circ f \circ ψ = p_{ϕ (i)} \circ ψ = ψ_{ϕ (i)}

and

p_{ϕ} \circ g \circ ψ = F (ϕ) \circ p_{i} \circ ψ = F (ϕ) \circ ψ_{i} = ψ_{ϕ (i)},

so there exists a unique morphism $𝜃 : X \to eq (f, g)$ with $p_{i} \circ ι \circ 𝜃 = ψ_{i}$ for all $i \in I$ . That is, $eq (f, g)$ satisfies the universal property of the limit. □

Definition 9.7.5. §

In a category $𝒞$ with a zero object $0$ , the zero morphism $0 : A \to B$ between objects $A, B \in Obj (𝒞)$ is the composition of $A \to 0 \to B$ of morphisms proscribed by the fact that $0$ is both initial and terminal.

Definition 9.7.6. §

Let $𝒞$ be a category with a zero object. Let $f : A \to B$ be a morphism in $𝒞$ , and let $0 : A \to B$ be the zero morphism.

a.: The kernel $\ker f$ of $f$ is the equalizer of $f$ and $0$ .
b.: The cokernel $coker f$ of $f$ is the coequalizer of $f$ and $0$ .

Examples 9.7.7. §

a.: In $R -mod$ , the categorical notions of kernel and cokernel agree with the usual ones.
b.: In $𝐆𝐩$ , kernel is the usual notion, and the cokernel of a group homomorphism $f : G \to H$ with the quotient of $H$ by the normal closure of $f (G)$ .

There are different notions of image and coimages in categories. We use the following.

Definition 9.7.8. §

Let $f : A \to B$ be a morphism in a category $𝒞$ .

a.: The image of $f$ is an object $im f$ of $𝒞$ together with a monomorphism $ι : im f \to B$ such that there exists a morphism $π : A \to im f$ such that $π \circ ι = f$ and such that if $g : C \to B$ us a monomorphism and $e : A \to C$ is a morphism such that $e \circ g = f$ , then there exists a unique morphism $ψ : im f \to C$ such that $g = ι \circ ψ$ .
b.: The coimage of $f$ is an object $coim f$ together with an epimorphism $π : A \to coim f$ such that there exists a morphism $ι : coim f \to B$ with $ι \circ π = f$ and such that if $e : A \to C$ is an epimorphism and $g : C \to B$ is a morphism such that $f = e \circ g$ , the there exists a unique morphism $𝜃 : coim f \to C$ such that $e = 𝜃 \circ f$ .

Remark 9.7.9. §

In the definition of the image of $f : A \to B$ , then $π$ is uniquely determined as $ι$ is a monomorphism and $f = ι \circ π$ . Moreover, if $g \circ ψ = ι$ as stated, then $g \circ ψ \circ π = ι \circ π = g \circ e$ , and $g$ is a monomorphism, so $ψ \circ π = e$ . That is the diagram

Universal factorization through an image. A full diagram description follows.

Diagram description: Universal factorization through an image

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

Row 1, from left to right: column 1: A; column 3: B.
Row 2, from left to right: column 2: image f.
Row 3, from left to right: column 2: C.

Arrows and lines:

An arrow from A to B, labelled f.
An arrow from A to image f, labelled pi.
A curved arrow from A to C, labelled e.
An arrow from image f to B, labelled iota.
An arrow from image f to C, labelled psi.
A curved arrow from C to B, labelled g.

commutes. Finally, note that $ψ$ is forced to be a monomorphism since $ι$ is. The analogous statements hold for the coimage.

Proposition 9.7.10. §

Let $f : A \to B$ be a morphism in a category $𝒞$ .

a.: If $𝒞$ admits equalizers, then the canonical morphism $π : A \to im f$ is an epimorphism.
b.: If $𝒞$ admits coequalizers, then the canonical morphism $ι : coim f \to B$ is a monomorphism.

Proof.

We prove only part a, as part b is proven similarly. Suppose that $α, β : im f \to D$ satisfy $α \circ π = β \circ π$ . Then by definition of the equalizer, there exists a unique morphism $ρ : A \to eq (α, β)$ such that for the canonical monomorphism $c : eq (α, β) \to im f$ , we have $π = c \circ ρ$ . On the other hand, consider the composite monomorphism $g = ι \circ c : eq (α, β) \to B$ , where $ι : im f \to B$ is the morphism of the image. Note that $g \circ ρ = f$ , so by definition of the image, there exists a unique morphism $d : im f \to eq (α, β)$ such that $g \circ d = ι$ . Then $ι \circ (c \circ d) = ι$ , so $c \circ d = id$ . On the other hand,

g \circ (d \circ c) = ι \circ c \circ d \circ c = ι \circ c = g .

As $g$ is a monomorphism, we have $d \circ c = id$ as well. □

Remark 9.7.11. §

If $𝒞$ has finite products, finite coproducts, equalizers, and coequalizers, we may also make the following definition of the image an coimage of a morphism $f : A \to B$ .

a.: The image of $f$ is the equalizer of the two morphisms $ι_{i} : B \to B ∐_{A} B$ .
b.: The coimage of $f$ is the coequalizer of the two projection morphisms $p_{i} : A \times_{B} A \to A$ .

This definition agrees with that already given if every morphism in $𝒞$ factors through an equalizer morphism and $𝒞$ admits finite limits and colimits. We omit the nontrivial proof.

Lemma 9.7.12. §

For any $f : A \to B$ in a category $𝒞$ that admits equalizers (or coequalizers) and for which $coim f$ and $im f$ exist, there is a unique morphism $u : coim f \to im f$ such that the composition

A \to_{}^{s} coim f \to_{}^{u} im f \to_{}^{t} B

of induced morphisms is $f$ .

Proof.

We suppose that $𝒞$ admits equalizers. By Proposition 9.7.10, the canonical morphism $π : A \to im f$ with $t \circ π = f$ is an epimorphism. By the definition of $coim f$ , there then exists a unique morphism $u : coim f \to im f$ such that $π = u \circ s$ . Then $t \circ u \circ s = t \circ π = f$ . If $v : coim f \to im f$ also satisfies $t \circ v \circ s = f$ , then $t \circ v \circ s = t \circ π$ , and $t : im f \to B$ is a monomorphism, so $v \circ s = π$ . Thus $u = v$ by the uniqueness of $u$ . □

Definition 9.7.13. §

We say that a morphism $f : A \to B$ in a category that admits an image and coimage of $f$ is strict if the induced morphism $coim f \to im f$ is an isomorphism.

Example 9.7.14. §

Every morphism in the category of $R$ -modules is strict.

9.8. Additive and abelian categories

Definition 9.8.1. §

An additive category $𝒞$ is a category with the following properties:

i.

for $A, B \in Obj (𝒞)$ , the set of morphisms ${Hom}_{𝒞} (A, B)$ in $𝒞$ has an abelian group law (addition) with the property that for any diagram

Additivity of composition. A full diagram description follows.

Diagram description: Additivity of composition

This diagram records the composable morphisms in the distributivity identity: h composed with (g subscript 1 plus g subscript 2) composed with f equals the sum of h composed with g subscript 1 composed with f and h composed with g subscript 2 composed with f.

Objects, listed by row and column:

Row 1, from left to right: column 1: A; column 2: B; column 3: C; column 4: D.

Arrows and lines:

An arrow from A to B, labelled f.
An arrow from B to C, labelled g subscript (1).
An arrow from B to C, labelled g subscript (2).
An arrow from C to D, labelled h.

in $𝒞$ , we have

h \circ (g_{1} + g_{2}) \circ f = h \circ g_{1} \circ f + h \circ g_{2} \circ f,

ii.

$𝒞$ has a zero object $0$ ,

iii.

$𝒞$ admits finite coproducts.

In an additive category $𝒞$ , there always exists the zero morphism is the identity element in the abelian group ${Hom}_{𝒞} (A, B)$ .

Examples 9.8.2. §

a.: The categories $𝐀𝐛$ and $R -mod$ are additive categories, with the usual addition of homomorphisms.
b.: The full subcategory $R -mod$ of finitely generated $R$ -modules is an additive category.

In an additive category, we denote the coproduct of two objects $A_{1}$ and $A_{2}$ by $A_{1} \oplus A_{2}$ .

Lemma 9.8.3. §

Finite products exist in an additive category, and there are natural isomorphisms

A_{1} ∐ A_{2} ≅ A_{1} \times A_{2}

for $A_{1}, A_{2} \in Obj (𝒞)$ . The resulting inclusion morphisms $ι_{i} : A_{i} \to A_{1} ∐ A_{2}$ and projection morphisms and $p_{i} : A_{1} ∐ A_{2} \to A_{i}$ obtained by viewing $A_{1} ∐ A_{2}$ as a product and coproduct, respectively, satisfy $p_{i} \circ ι_{i} = {id}_{A_{i}}$ and $p_{i} \circ ι_{j} = 0$ for $i \neq j$ , while

ι_{1} \circ p_{1} + ι_{2} \circ p_{2} = {id}_{A_{1} ∐ A_{2}} .

Proof.

We have morphisms $ι_{i} : A_{i} \to A_{1} ∐ A_{2}$ by definition. We also have maps $p_{i} : A_{1} ∐ A_{2} \to A_{i}$ defined by

p_{i} \circ ι_{j} = {\begin{matrix} {id}_{A_{i}} & if i = j \\ 0 & if i \neq j \end{matrix}

and the universal property of the coproduct. We then have

(ι_{1} \circ p_{1} + ι_{2} \circ p_{2}) \circ ι_{i} = ι_{i},

and hence $ι_{1} \circ p_{1} + ι_{2} \circ p_{2} = {id}_{A_{1} ∐ A_{2}}$ , again by the universal property.

Given an object $B \in Obj (𝒞)$ and morphisms $g_{i} : B \to A_{i}$ , we then have a morphism

ψ = ι_{1} \circ g_{1} + ι_{2} \circ g_{2} : B \to A_{1} ∐ A_{2},

which is unique such that

p_{i} \circ ψ = g_{i} .

Hence $A_{1} ∐ A_{2}$ satisfies the universal property of the product. □

Definition 9.8.4. §

An object $A$ in an additive category $𝒞$ together with objects $A_{i}$ , inclusion morphisms $ι_{i} : A_{i} \to A$ , and projection morphisms $p_{i} : A \to A_{i}$ for $i \in {1, 2}$ for which $p_{i} \circ ι_{j}$ is zero if $i \neq j$ and ${id}_{A_{i}}$ if $i = j$ and for which $ι_{1} \circ p_{1} + ι_{2} \circ p_{2} = {id}_{A}$ is called a biproduct of the objects $A_{1}$ and $A_{2}$ , and we write it as $A_{1} \oplus A_{2}$ .

The notion of a biproduct allows us to reinterpret addition in an additive category. First, note the following definitions.

Definition 9.8.5. §

Let $A$ be an object in an additive category $𝒞$ .

a.: The diagonal morphism $Δ_{A} : A \to A \oplus A$ in $𝒞$ is the unique morphism induced by two copies of ${id}_{A} : A \to A$ and the universal property of the product.
b.: The codiagonal morphism $\nabla_{A} : A \oplus A \to A$ in $𝒞$ is the unique morphisms induced by two copies of ${id}_{A} : A \to A$ and the universal property of the coproduct.

Definition 9.8.6. §

Let $𝒞$ be an additive category, and let $f_{1} : A_{1} \to B_{1}$ and $f_{2} : A_{2} \to B_{2}$ be morphisms in $𝒞$ . The biproduct, or direct sum, $f_{1} \oplus f_{2}$ of the maps $f_{1}$ and $f_{2}$ is the morphism $A_{1} \oplus A_{2} \to B_{1} \oplus B_{2}$ induced as the (morphism defined by the universal property of the) coproduct of the composite maps $A_{i} \to B_{i} \to B_{1} \oplus B_{2}$ , the latter morphisms being inclusions.

Remark 9.8.7. §

Equivalently, the direct sum of $f_{1}$ and $f_{2}$ as in Definition 9.8.6 is induced as the product of the composite maps $A_{1} \oplus A_{2} \to A_{i} \to B_{i}$ , the initial morphisms being projections.

Of course, we could make these definitions in an arbitrary category using products and coproducts.

Lemma 9.8.8. §

Let $f, g : A \to B$ be two morphisms in an additive category $𝒞$ . Then we have

f + g = \nabla_{B} \circ (f \oplus g) \circ Δ_{A} .

Proof.

Let $ι_{i}^{A}$ and $p_{i}^{A}$ respectively denote the inclusion maps and projection maps for the biproduct $A \oplus A$ , and similarly for $B$ . We have

\begin{matrix} \nabla_{B} \circ (f \oplus g) \circ Δ_{A} = \nabla \circ (f \oplus g) \circ (ι_{1}^{A} \circ p_{1}^{A} + ι_{2}^{A} \circ p_{2}^{A}) \circ Δ_{A} \\ = \nabla_{B} \circ (f \oplus g) \circ ι_{1}^{A} \circ p_{1}^{A} \circ Δ_{A} + \nabla_{B} \circ (f \oplus g) \circ ι_{2}^{A} \circ p_{2}^{A} \circ Δ_{A} . \end{matrix}

Taking the first term without loss of generality, we have

\nabla_{B} \circ ((f \oplus g) \circ ι_{1}^{A}) \circ (p_{1}^{A} \circ Δ_{A}) = \nabla_{B} \circ (ι_{1}^{B} \circ f) \circ {id}_{A} = {id}_{B} \circ f = f .

□

Definition 9.8.9. §

A functor $F : 𝒞 \to 𝒟$ between additive categories is called additive if for each $A, B \in Obj (𝒞)$ , the map

{Hom}_{𝒞} (A, B) \to {Hom}_{𝒟} (F (A), F (B))

is a group homomorphism.

Example 9.8.10. §

Let $𝒞$ be an additive category. Then for any $A \in Obj (𝒞)$ , the functors $h^{A}$ and $h_{A}$ may be considered as functors to $𝐀𝐛$ , rather than $𝐒𝐞𝐭$ . The resulting functors are additive.

Lemma 9.8.11. §

A functor $F : 𝒞 \to 𝒟$ of additive categories is additive if and only if $F$ preserves biproducts, which is to say that the natural morphisms $F (A_{1}) \oplus F (A_{2}) \to F (A_{1} \oplus A_{2})$ and $F (A_{1} \oplus A_{2}) \to F (A_{1}) \oplus F (A_{2})$ are inverse isomorphisms for all objects $A_{1}, A_{2}$ in $𝒞$ .

Proof.

Suppose first that $F$ is an additive functor. Note that $F (ι_{i} \circ p_{i}) = {id}_{F (A_{i})}$ and $F (ι_{i} \circ p_{j}) = F (0)$ for $i \neq j$ , but $F (0) = 0$ by additivity of $F$ . Again by additivity of $F$ , we have

F (ι_{1}) \circ F (p_{1}) + F (ι_{2}) \circ F (p_{2}) = F ({id}_{A_{1} \oplus A_{2}}) = {id}_{F (A_{1} \oplus A_{2})} .

It follows that $F (A_{1} \oplus A_{2})$ is a biproduct of $F (A_{1})$ and $F (A_{2})$ in $𝒟$ , so in particular it is a coproduct.

On the other hand, if $F$ preserves biproducts and $f, g : A \to B$ are morphisms in $𝒞$ , then it is easy to see that $F (f \oplus g) = F (f) \oplus F (g)$ , and Lemma 9.8.8 tells us that

F (f + g) = F (\nabla_{B} \circ (f \oplus g) \circ Δ_{A}) = \nabla_{F (B)} \circ (F (f) \oplus F (g)) \circ Δ_{F (A)} = F (f) + F (g) .

□

Corollary 9.8.12. §

Let $F : 𝒞 \to 𝒟$ be a fully faithful functor of additive categories. Then $F$ is an additive functor.

Remark 9.8.13. §

For additive functors, we may consider a finer notion of representability than we previously studied. If $F : 𝒞 \to 𝐀𝐛$ is an additive contravariant (resp., covariant) functor of additive categories, then we may consider it to be representable if there exists an object $X \in Obj (𝒞)$ and a natural isomorphism $η : h^{X} ⇝ F$ (resp., $η : h_{X} ⇝ F$ ). In this case, the morphisms $η_{A}$ for $A \in Obj (𝒞)$ will be isomorphisms of groups.

Lemma 9.8.14. §

A morphism in an additive category that admits kernels is a monomorphism if and only if it has zero kernel. A morphism in an additive category that admits cokernels is an epimorphism if and only if it has zero cokernel.

Proof.

Let $f : A \to B$ be a monomorphism, and let $h : \ker f \to A$ be the induced morphism. Since $f \circ h = 0$ by definition of the kernel, we have $h = 0$ , as $f$ is a monomorphism. This forces $\ker f$ to be $0$ , since $h$ factors through $0$ . (Or, one could just apply Lemma 9.7.3.) On the other hand, suppose that $f$ has trivial kernel, and let $g, h : C \to A$ be maps with $f \circ g = f \circ h$ . Then $f \circ (g - h) = 0$ , and by universal property of the kernel, $g - h$ factors through $0$ , i.e., is $0$ .

The proof for cokernels is similar, or is the result on kernels in the opposite (additive) category. □

Proposition 9.8.15. §

Let $𝒞$ be an additive category that admits kernels and cokernels. Let $f : A \to B$ be a morphism in $𝒞$ . Then

im f ≅ \ker (B \to coker f)

and

coim 𝑓≅ coker (\ker f \to A) .

Proof.

We prove the first isomorphism. Let $g : B \to coker f$ . By Yoneda’s lemma, it suffices to show that $h^{im f}$ and $h^{\ker g}$ are naturally isomorphic. For $C \in Obj (𝒞)$ , we have a map

{Hom}_{𝒞} (C, im f) \to_{}^{\sim} {α : C \to B ∣ ι_{1} \circ α = ι_{2} \circ α}

that takes a morphism $C \to im f$ and composes it with the morphism $im f \to B$ given by definition of the equalizer of the maps $ι_{i} : B \to B ∐_{A} B$ . It is a bijection by the universal property of the equalizer.

For any $D \in Obj (𝒞)$ and morphisms $ϕ_{1}, ϕ_{2} : B \to D$ such that $ϕ_{1} \circ f = ϕ_{2} \circ f$ , note that there exists a unique morphism $k : B ∐_{A} B \to D$ with $ϕ_{i} = k \circ ι_{i}$ . Any $α : C \to B$ such that $ι_{1} \circ α = ι_{2} \circ α$ then satisfies $ϕ_{1} \circ α = ϕ_{2} \circ α$ for any such $ϕ_{i} : B \to D$ and any $D$ . On the other hand, note that the $ι_{i}$ themselves satisfy the property that $ι_{1} \circ f = ι_{2} \circ f$ and are morphisms $ι_{i} : B \to D$ with $D = B ∐_{A} B$ . In other words, we have

\begin{matrix} {α : C \to B ∣ ι_{1} \circ α = ι_{2} \circ α} \\ = {α : C \to B ∣ ϕ_{1} \circ α = ϕ_{2} \circ α if ϕ_{1} \circ f = ϕ_{2} \circ f for some ϕ_{1}, ϕ_{2} : B \to D (for some D \in Obj (𝒞))} . \end{matrix}

Now, we are in an additive category, so this equals

{α : C \to B ∣ ϕ \circ α = 0 if ϕ \circ f = 0 for some ϕ : B \to D} .

(9.8.1)

By the universal property of $coker f$ , for any $ϕ : B \to D$ with $ϕ \circ f = 0$ , there is a morphism $j : coker f \to D$ with $j \circ g = ϕ$ . If $g \circ α = 0$ , then $ϕ \circ α = j \circ g \circ α = 0$ , and this works for any $ϕ : B \to D$ with $ϕ \circ f = 0$ . On the other hand, $g$ itself satisfies $g \circ f = 0$ so is such a $ϕ$ . It follows that the set in (9.8.1) equals

{α : C \to B ∣ g \circ α = 0} .

By the universal property of $\ker g$ , this is in bijection with ${Hom}_{𝒞} (C, \ker g)$ , taking an $α$ in the set to the unique morphism to $\ker g$ through which it factors. Clearly, the composition of these bijections is natural in $C$ , so we have the desired natural isomorphism. □

Definition 9.8.16. §

An abelian category is an additive category $𝒞$ in which

i.: every morphism in $𝒞$ admits a kernel and a cokernel and
ii.: every morphism in $𝒞$ is strict.

Examples 9.8.17. §

a.: The category $R -mod$ is abelian.
b.: The full subcategory $𝒞$ of $R -mod$ of finitely generated $R$ -submodules is not necessarily abelian. E.g., when $R$ is commutative and non-noetherian, we can take $I$ to be an ideal of $R$ that is not finitely generated, and so the kernel of $R \to R ∕ I$ is not in $𝒞$ .

Remark 9.8.18. §

Note that if $𝒞$ is an abelian category, then so is $𝒞^{op}$ . The roles of mono- and epimorphisms, kernels and cokernels, and images and coimages switch in $𝒞$ and $𝒞^{op}$ .

Proposition 9.8.19. §

The functor category $𝐅𝐮𝐧𝐜 (𝒞, 𝒟)$ from a small category $𝒞$ to an abelian category $𝒟$ is abelian.

Proof.

We sketch the proof. First, note that it is additive: we have the zero functor which sends all objects to the zero object and all morphisms to the zero (identity) morphism of the zero object, and if $F, G : 𝒞 \to 𝒟$ are functors, then $F \oplus G$ is given by $(F \oplus G) (C) = F (C) \oplus G (C)$ and $(F \oplus G) (f) = F (f) \oplus G (f)$ for $f : A \to B$ in $C$ . This can be used to define the addition on morphisms (i.e., natural transformations) as before.

Next, the kernel of a natural transformation $η : F ⇝ G$ is defined by $(\ker η) (C) = \ker η_{C}$ and $(\ker η) (f)$ for $f : A \to B$ is the kernel of the induced morphism $\ker η_{A} \to \ker η_{B}$ . The cokernel is defined similarly. Note that

{(coim η)}_{A} ≅ coker {(\ker η ⇝ η)}_{A} ≅ coker (\ker η_{A} ⇝ η_{A}) ≅ coim η_{A},

and similarly for images. Finally, since $𝒟$ is abelian, the natural map $coim η ⇝ im η$ is an isomorphism $coim η_{A} \to im η_{A}$ on objects $A$ in $𝒞$ , hence has a natural inverse determined by the inverses of these morphisms. □

Terminology 9.8.20. §

In an abelian category $𝒞$ , we typically refer to a coproduct (when it exists) as a direct sum, and we write $\oplus_{i \in I} A_{i}$ in place of $∐_{i \in I} A_{i}$ .