Homological Algebra — Chapter 1 Category Theory

Chapter 1
Category Theory

1.1. 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 1.1.1. §

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 map

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

for each $A, B, C \in Obj (𝒞)$ , called composition, where, for $f : A \to B$ and $g : B \to C$ , their composition is denoted $g \circ f$ ,

such 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 1.1.2. §

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

Definition 1.1.3. §

We say that a category $𝒞$ is locally small if ${Hom}_{𝒞} (A, B)$ is a set for all $A, B \in Obj (𝒞)$ .

Every example of a category we give will be locally small.

Examples 1.1.4. §

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.

Definition 1.1.5. §

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 1.1.6. §

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

Example 1.1.7. §

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 1.1.8. §

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 1.1.9. §

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 1.1.10. §

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 $i = j$ ).

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

Definition 1.1.11. §

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 1.1.12. §

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 1.1.13. §

Given a category $𝒞$ , its morphism category $Mor (𝒞)$ is the category with objects the morphisms in $𝒞$ and morphisms $f \to g$ for $f : A \to B$ and $g : A^{'} \to B^{'}$ in $𝒞$ the pairs $(α, β)$ of morphisms $α : A \to A^{'}$ and $β : B \to B^{'}$ in $𝒞$ with $β \circ f = g \circ α$ .

Example 1.1.14. §

Let $G$ be a monoid, which is to say a set with an associative binary operation and a two-sided identity element with respect to the operation. Then $G$ is a category with one object, morphisms equal to its elements, 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 1.1.15. §

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 1.1.16. §

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.

Example 1.1.17. §

Under the correspondence between monoids and single-object categories, groups correspond to exactly those single-object categories in which every morphism is an isomorphism.

Definition 1.1.18. §

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 1.1.19. §

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 1.1.20. §

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 1.1.21. §

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 1.1.20. □

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

Definition 1.1.22. §

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 $π : C \to B$ .

1.2. Functors and natural transformations

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 1.2.1. §

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 1.2.2. §

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.

For any category $𝒞$ , we have a contravariant functor $op : 𝒞^{op} \to 𝒞$ (as well as $𝒞 \to 𝒞^{op}$ ) given by the identity on objects and the map on morphisms that takes $f : B \to A$ to $f^{op} : A \to B$ . The composition $F \circ op$ for a contravariant functor $F$ is a covariant functor $𝒞^{op} \to 𝒟$ .

e.

Given an object $A \in Obj (𝒞)$ for some category $𝒞$ , we can define a functor $h_{A} : 𝒞 \to Sets$ by

h_{A} (B) = {Hom}_{𝒞} (A, B)

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

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

for all $f : A \to B$ .

f.

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 1.2.3. §

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

Remark 1.2.4. §

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 1.2.5. §

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

Definition 1.2.6. §

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 1.2.7. §

To give a functor from $I$ as in Example 1.1.9 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 1.2.8. §

To give a functor from $I$ as in Example 1.1.10 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 $g \circ f = {id}_{A}$ .

Remark 1.2.9. §

A morphism $f \to g$ in $Mor (𝒞)$ for a category $𝒞$ may be thought of as a commutative diagram

A morphism in the category of morphisms. A full diagram description follows.

Diagram description: A morphism in the category of morphisms

The structural squares and triangles displayed here commute.

Objects, listed by row and column:

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

Arrows and lines:

An arrow from A to B, labelled f.
An arrow from A to A prime, labelled alpha.
An arrow from B to B prime, labelled beta.
An arrow from A prime to B prime, labelled g.

in $𝒞$ .

Definition 1.2.10. §

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.

Remark 1.2.11. §

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 1.2.12. §

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 1.2.13. §

The inclusion functor attached to a subcategory is always faithful.

Remark 1.2.14. §

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 1.2.15. §

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.

Clearly, a functor always takes isomorphisms to isomorphisms. Of course, a fully faithful functor $F$ preserves notions of monomorphism, epimorphism, and isomorphism (in that $f$ has one of these properties if and only if $F (f)$ has the same property). The reader may also quickly check the following.

Lemma 1.2.16. §

Let $F : 𝒞 \to 𝒟$ be a functor, and let $f : A \to B$ be a morphism in $𝒞$ . If $F$ is faithful and $F (f)$ is a monomorphism (resp., epimorphism) then $f$ is a monomorphism (resp., epimorphism).

Definition 1.2.17. §

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.

Remark 1.2.18. §

Given two categories $𝒞$ and $𝒟$ with $𝒞$ small and defining composition of natural transformations in the obvious way, we get a functor category $𝐅𝐮𝐧𝐜 (𝒞, 𝒟)$ with objects the functors $𝒞 \to 𝒟$ and morphisms the natural transformations between functors.

Example 1.2.19. §

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

Definition 1.2.20. §

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 1.2.21. §

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 1.2.22. §

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 $η : F \to G$ .

1.3. The Yoneda embedding

Definition 1.3.1. §

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 1.3.2. §

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

Theorem 1.3.3. §

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 1.3.4. §

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

Corollary 1.3.5 (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 actually require the following strengthening of Theorem 1.3.3.

Theorem 1.3.6 (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. □

1.4. Limits and colimits

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

Notation 1.4.1. §

We write $i \in I$ to denote, more simply, that $i \in Obj (I)$ .

Definition 1.4.2. §

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 1.4.3. §

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 1.4.4. §

The universal property of the limit of a functor $F$ as in Definition 1.4.2 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 1.4.5. §

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 1.4.6. §

Lemma 1.4.5 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 1.4.7. §

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 1.4.8. §

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 1.4.9. §

Given a commutative diagram in a category $𝒞$ , it arises by definition 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 1.4.10. §

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).

(1.4.1)

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

Remark 1.4.11. §

The pullback of (1.4.1) 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 1.4.12. §

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 1.4.13. §

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 $colim F \in Obj (𝒞)$ 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 $ψ_{j} \circ κ = ψ_{i}$ 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 1.4.14. §

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

Remark 1.4.15. §

The properties of the colimit expressed in Definition 1.4.13 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 1.4.5, which again tells us that we may speak of “the” colimit.

Lemma 1.4.16. §

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 1.4.17. §

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

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

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

Definition 1.4.18. §

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 1.4.19. §

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 CRings of commutative rings $R_{1}$ and $R_{2}$ is the tensor product $R_{1} \otimes R_{2}$ .

Remark 1.4.20. §

Examples 1.4.19(a-d) generalize directly to arbitrary collections of objects.

Remark 1.4.21. §

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

Definition 1.4.22. §

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).

(1.4.2)

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

Remark 1.4.23. §

The pushout of the diagram (1.4.2) 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 1.4.24. §

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 1.4.25. §

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 1.4.26. §

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

Definition 1.4.27. §

A category is called complete if it admits all limits.

Example 1.4.28. §

The category of finite sets is not complete.

Definition 1.4.29. §

A category is called cocomplete if it admits all colimits.

Remark 1.4.30. §

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

Proposition 1.4.31. §

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 $F (ϕ) (a_{i}) = a_{j}$ . □

The reader will easily verify the following.

Proposition 1.4.32. §

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

Corollary 1.4.33. §

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 $𝐅𝐮𝐧𝐜 (𝒞^{op}, 𝐒𝐞𝐭)$ .

We explain some restricted and successively more general notions of limit and colimit.

Definition 1.4.34. §

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 object which is both initial and terminal is called a zero object.

Remarks 1.4.35. §

a.: Consider the empty functor $F : \emptyset \to 𝒞$ . Then $\lim F$ is a terminal object, and $colim F$ is an initial object, if they exist.
b.: Terminal and initial objects are unique up to unique isomorphism when defined.

We provide some examples.

Examples 1.4.36. §

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 1.4.37. §

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
$colim F = F (j) .$

Definition 1.4.38. §

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 1.4.39. §

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.

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

Definition 1.4.40. §

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 1.4.41. §

Let $I$ be a category such that $I^{op}$ is directed. 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 1.4.42. §

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 ${lim_{\to}}_{} F$ ).

It turns out that it is useful to generalize the notion of a directed system slightly in the context of category theory, keeping the requirement of being directed but weakening the requirement of the category arising from a partial ordering.

Definition 1.4.43. §

A small category $I$ is said to be filtered if it is nonempty and the following hold.

i.: For every pair $(i, j)$ of objects in $I$ , there exists an object $k$ and morphisms $i \to k$ and $j \to k$ in $I$ .
ii.: For every two morphisms $κ, κ^{'} : i \to j$ in $I$ , there exists a morphism $λ : j \to k$ in $I$ such that $λ \circ κ = λ \circ κ^{'}$ .

Definition 1.4.44. §

A small category is said to be cofiltered if its opposite category is filtered.

Example 1.4.45. §

Any small category with a terminal (resp., initial) object is filtered (resp., cofiltered).

We are typically interested in cofiltered limits and filtered colimits, of which inverse limits and direct limits are respectively special cases. We denote such limits as we do inverse and direct limits. The axiom of choice, together with Lemmas 1.4.5 and 1.4.16, allows us to define functors as in the following definition.

Definition 1.4.46. §

Let $𝒞$ be a category and $I$ a small category. Suppose that $𝒞$ admits limits (resp., colimits) from $I$ . Then the limit (resp., colimit) functor

\lim : 𝐅𝐮𝐧𝐜 (I, 𝒞) \to 𝒞 (resp., colim : 𝐅𝐮𝐧𝐜 (I, 𝒞) \to 𝒞)

is any functor that takes a functor $F : I \to 𝒞$ to a limit (resp., colimit) of $F$ and a natural transformation $η : F ⇝ G$ to the unique morphism

\lim η : \lim F \to \lim G (resp., colim η : colim F \to colim G)

given by the universal property. When $I$ is cofiltered (resp., filtered), the limit (resp., colimit) functor is denoted ${lim_{\leftarrow}}_{}$ (resp., ${lim_{\to}}_{}$ ).

1.5. Adjoint functors

Definition 1.5.1. §

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

η_{(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 $η_{(C, D)}$ 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 1.5.2. §

To say that $η$ is a natural transformation in Definition 1.5.1 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 1.5.3. §

Adjointness is a weak form of the property of being mutually inverse. I.e., if $G$ is a left and right inverse to $F$ (on objects and morphisms), then we have

{Hom}_{𝒟} (F (C), D) ≅ {Hom}_{𝒞} (G \circ F (C), G (D)) = {Hom}_{𝒞} (C, G (D)),

so $F$ is left adjoint to $G$ , and similarly, $F$ is right adjoint to $G$ .

The following provides a standard example of adjoint functors.

Example 1.5.4. §

The forgetful functor $𝐆𝐩 \to 𝐒𝐞𝐭$ is right adjoint to the functor $𝐒𝐞𝐭 \to 𝐆𝐩$ taking a set to the free group on it. That is, for any set $X$ and group $𝒢$ , there is a bijection of sets

Hom (F_{X}, 𝒢) \to Maps (X, 𝒢),

where $F_{X}$ denotes the free group on $X$ .

Proposition 1.5.5. §

Let $S$ be a set. The functor $h_{S} : 𝐒𝐞𝐭 \to 𝐒𝐞𝐭$ is right adjoint to the functor $t_{S} : 𝐒𝐞𝐭 \to 𝐒𝐞𝐭$ given by $t_{S} (T) = T \times S$ and $t_{S} (f) = f \times {id}_{S}$ for any sets $T, T^{'}$ and $f : T \to T^{'}$ .

Proof.

We define our bijections by

\begin{array}{rcl} τ_{T, U} : Maps (T \times S, U) \to Maps (T, Maps (S, U)) \\ τ_{T, U} (f) = (t \mapsto (s \mapsto f (t, s))) . \end{array}

We leave the verification of naturality to the reader. □

Later, we will treat what is perhaps the most standard example of adjointness: that of $Hom$ and $\otimes$ in categories of modules.

Proposition 1.5.6. §

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 1.5.7. §

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 1.5.8. §

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.

Notation 1.5.9. §

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 1.5.10. §

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 1.5.11. §

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. □

1.6. Representable functors

Definition 1.6.1. §

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 1.6.1 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 1.6.2. §

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 1.6.3. §

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 1.6.4. §

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 1.6.3.

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 1.6.5. §

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 1.6.6. §

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 1.6.7. §

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 1.6.8. §

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 1.6.9. §

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 1.6.10. §

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 $𝒞$ .

1.7. Equalizers and images

Definition 1.7.1. §

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.

(1.7.1)

be a diagram in $𝒞$ .

a.: The limit $eq (f, g)$ of the diagram (1.7.1), when it exists, is called its equalizer.
b.: The colimit $coeq (f, g)$ of (1.7.1) 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 1.7.2. §

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 1.7.3. §

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. □

We begin with the notions of image and coimage in an arbitrary category.

Definition 1.7.4. §

Let $f : A \to B$ be a morphism in a category that has finite products, finite coproducts, equalizers, and coequalizers.

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 $p_{i} : A \times_{B} A \to A$ .

Remark 1.7.5. §

The image of a morphism $f : A \to B$ fits in a diagram

The image of a morphism. A full diagram description follows.

Diagram description: The image of a morphism

The outer pushout square commutes. The central image object receives the factorization of f from A and maps to the two copies of B; these factorization triangles 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 1: B; column 3: B coproduct subscript (A) B.

Arrows and lines:

An arrow from A to B (row 1, column 3), labelled f.
An arrow from A to B (row 3, column 1), labelled f.
An arrow from A to image f, without a label.
An arrow from B (row 1, column 3) to B coproduct subscript (A) B, labelled iota subscript (1).
An arrow from image f to B (row 1, column 3), without a label.
An arrow from image f to B (row 3, column 1), without a label.
An arrow from B (row 3, column 1) to B coproduct subscript (A) B, labelled iota subscript (2).

(1.7.2)

where the morphism $A \to im f$ is induced by the universal property of $im f$ and the morphism $im f \to B$ are identical to each other. Note that $im f \to B$ is a monomorphism, since it is an equalizer. Dually, we also have an induced epimorphism $coim f \to B$ .

Example 1.7.6. §

We check that the definition of $im f$ agrees with the usual notion in the category of $R$ -modules. We claim that ${f (a) ∣ a \in A}$ and its inclusion in $B$ is the equalizer of the diagram

The image as an equalizer. A full diagram description follows.

Diagram description: The image as an equalizer

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: B coproduct subscript (A) B.

Arrows and lines:

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

Since

B ∐_{A} 𝐵≅ \frac{B \oplus B}{{(f (a), 0) - (0, f (a)) ∣ a \in A}},

the claim follows from Example 1.7.2(b).

Lemma 1.7.7. §

For any $f : A \to B$ in a category that admits equalizers and coequalizers 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.

Consider the diagram

Factorization through coimage and image. A full diagram description follows.

Diagram description: Factorization through coimage and image

This diagram describes the factorization of f through its coimage and image, and the equalizer and coequalizer maps that define them. The parallel arrows need not be equal; their relevant composites through f are equal.

Objects, listed by row and column:

Row 1, from left to right: column 1: A times subscript (B) A; column 2: A; column 3: B; column 4: B coproduct subscript (A) B.
Row 2, from left to right: column 2: coimage f; column 3: image f.

Arrows and lines:

An arrow from A times subscript (B) A to A, labelled p subscript (2).
An arrow from A times subscript (B) A to A, labelled p subscript (1).
An arrow from A to B, labelled f.
An arrow from A to coimage f, labelled s.
An arrow from A to image f, labelled r.
An arrow from B to B coproduct subscript (A) B, labelled iota subscript (2).
An arrow from B to B coproduct subscript (A) B, labelled iota subscript (1).
An arrow from image f to B, labelled t.

(1.7.3)

where the diagonal map $r$ is from (1.7.2). We have that $f \circ p_{1} = f \circ p_{2}$ , so

t \circ r \circ p_{1} = t \circ r \circ p_{2}

We know that $t$ is a monomorphism since the image is an equalizer, so $r \circ p_{1} = r \circ p_{2}$ . By the universal property of the coimage, it follows that there exists a unique map $u : coim f \to im f$ making the diagram (1.7.3) commute. □

Definition 1.7.8. §

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

Example 1.7.9. §

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