Category Theory Preliminaries

Categories

Definition 318.1(Category).

A category $\text{C}$ consists of

• a class $\text{Obj}(\text{C})$ of objects of this category; and
• for every two objects $A,B$ of $\text{C}$, a set ${\text{Hom}}_{\text{C}}(A,B)$ of morphisms, with the properties listed below.

Properties of morphisms:

1. For every object $A$ of $\text{C}$, there exists a morphism $1_A\in {\text{Hom}}_{\text{C}}(A,A)$, called the identity on $A$.
2. One can compose morphisms: two morphisms $f\in {\text{Hom}}_{\text{C}}(A,B)$ and $g\in {\text{Hom}}_{\text{C}}(B,C)$ determine a morphism $gf\in {\text{Hom}}_{\text{C}}(A,C)$. That is, for every triple of objects $A,B,C$ of $\text{C}$, there is a function (of sets) ${\text{Hom}}_{\text{C}}(A,B)\times {\text{Hom}}_{\text{C}}(B,C)\to {\text{Hom}}_{\text{C}}(A,C)$, and the image of the pair $(f,g)$ is denoted by $gf$.
3. This ‘composition law’ is associative: if $f\in {\text{Hom}}_{\text{C}}(A,B)$, $g\in {\text{Hom}}_{\text{C}}(B,C)$, and $h\in {\text{Hom}}_{\text{C}}(C,D)$, then $(hg)f=h(gf)$.
4. The identity morphisms are identities with respect to composition: for all $f\in {\text{Hom}}_{\text{C}}(A,B)$, we have $f{1}_A=f$, $1_{B}f=f$.

A category is called concrete if the objects of the category are structured sets and the arrows of the category are (certain) functions.

Example 318.2(Concrete categories).

• Groups and group homomorphisms
• $R$-modules and $R$-module homomorphisms
• Vector spaces and linear transformations
• Topological spaces and continuous maps
• Posets and monotone functions

The strength of category theory lies in its abstraction: objects do not have to be sets, and morphisms do not have to be functions.

Example 318.3(Finite categories).

Example 318.4(The category Rel).

Take sets as objects and take binary relations as arrows. That is, an arrow $f:A\to B$ is an arbitrary subset $f\subseteq A\times B$. The identity arrow on a set $A$ is the identity relation. Given $R\subseteq A\times B$ and $S\subseteq B\times C$, define composition $S\circ R$ by

$$(a,c)\in S\circ R⟺\exists b.(a,b)\in R\&(b,c)\in S.$$

Example 318.5(Preorders).

A preorder is a set $P$ equipped with a binary relation $p\le q$ thata is both reflexive and transitive: $a\le a$, and if $a\le b$ and $b\le c$, then $a\le c$ (what differentiates a preorder from a poset is the lack of antisymmetry). Any preorder $P$ can be regarded as a category by taking the objects to be the elements of $P$ and taking a unique arrow, $a\to b$ if and only if $a\le b$. The reflexive and transitive conditions on $\le$ ensure that this is indeed a category. Note that $\text{Hom}(a,b)$ for any $a,b\in P$ is a singleton if $a\le b$, and empty otherwise.

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Example 318.6(Slice categories).

Let $\text{C}$ be a category, and fix an object $A$ of $\text{C}$. Define a new category ${\text{C}}_A$ as follows:

• $\text{Obj}({\text{C}}_A)$ consists of all morphisms from any object of $\text{C}$ to $A$; thus, objects of ${\text{C}}_A$ are morphisms of the type

• morphisms

  are commutative diagrams

Example 318.7(Example).

Let $\text{C}$ be a category. Let $A,B\in \text{Obj}(\text{C})$. Define a new category ${\text{C}}_{A,B}$ like so:

• $\text{Obj}({\text{C}}_{A,B})$ consists of diagrams

  in $\text{C}$, denoted by $(Z,f,g)$; and

• morphisms

  are commutative diagrams

As in example 3.5, the identities are inherited from the identities in $\text{C}$: for $(Z,f,g)$ in ${\text{C}}_{A,B}$, the identity $1_{(Z,f,g)}$ corresponds to the diagram

The composition $(Z_1,f_1,g_1)\stackrel{\sigma }{\to }(Z_2,f_2,g_2)\stackrel{\tau }{\to }(Z_3,f_3,g_3)$ is achieved by combining the diagrams of $\sigma$ and $\tau$ in the following manner

and since $\text{C}$ is a category, it follows that the diagram obtained by removing $Z_2$, i.e,

commutes. Associativity immediately follows from the fact that composition is associative in $\text{C}$.

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Morphisms

Isomorphisms and automorphisms

Definition 318.8(Isomorphism).

A morphism $f\in {\text{Hom}}_{\text{C}}(A,B)$ is an isomorphism if it has a (two-sided) inverse under composition: that is, if $\exists g\in {\text{Hom}}_{\text{C}}(B,A)$ such that

$$gf=1_A,fg=1_B.$$

Groupoids

A category in which every morphism is an isomorphism is called a groupoid. If the preorder in Example 5 is symmetric, the category constructed is a groupoid (every morphism now has an inverse).

Proposition 318.9(Proposition).

The inverse of an isomorphism is unique.

To hammer the point home, if $f$ is a morphism with a left-inverse $g_1$ and a right inverse $g_2$, then $f$ is necessarily an isomorphism, $g_1=g_2$, and this morphism is the (unique) inverse of $f$.

More utterly standard stuff:

Proposition 318.10(Proposition).

• Each identity $1_A$ is an isomorphism and its own inverse.
• If $f$ is an isomorphism, then $f^{-1}$ is an isomorphism and further $(f^{-1}{)}^{-1}=f$.
• If $f\in {\text{Hom}}_{\text{C}}(A,B)$, $g\in {\text{Hom}}_{\text{C}}(B,C)$ are isomorphisms, then $gf$ is an isomorphism and $(gf{)}^{-1}=f^{-1}{g}^{-1}$.

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A morphism of an object $A$ of a category $\text{C}$ to itself is called an endomorphism. ${\text{Hom}}_{\text{C}}(A,A)$ is denoted by ${\text{End}}_{\text{C}}(A)$.

Definition 318.11(Automorphism).

An automorphism of an object $A$ of a category $\text{C}$ is an isomorphism form $A$ to itself. The set of automorphisms of $A$ is denoted by ${\text{Aut}}_{\text{C}}(A)$; it is a subset of ${\text{End}}_{\text{C}}(A)$.

From Proposition 10, we can infer than ${\text{Aut}}_{\text{C}}(A)$ is a group for all objects $A$ of all categories $\text{C}$.

Monomorphisms and epimorphisms

Note that defining qualities of morphisms by their actions on ‘elements’ (as we did in $\text{Set}$) is not an option here, because objects of an arbitrary category do not (in general) have ‘elements’. However, recall that properties of morphisms in $\text{Set}$ such as injectivity and surjectivity did have alternative formulations which did not reference ‘elements’ at all (left cancellable and right cancellable functions, respectively). It turns out that these formulations of these notions do transfer nicely into the categorical setting.

Definition 318.12(Monomorphism).

Let $\text{C}$ be a category. A morphism $f\in {\text{Hom}}_{\text{C}}(A,B)$ is a monomorphism if the following holds: for all objects $Z$ of $\text{C}$ and all morphisms $\alpha ,{\alpha }^{\prime }\in {\text{Hom}}_{\text{C}}(Z,A)$,

$$f\circ \alpha =f\circ {\alpha }^{\prime }⟹\alpha ={\alpha }^{\prime }.$$

In other words, $f$ is a monomorphism if it is left cancellable.

Definition 318.13(Epimorphism).

Let $\text{C}$ be a category. A morphism $f\in {\text{Hom}}_{\text{C}}(A,B)$ is an epimorphism if the following holds: for all objects $Z$ of $\text{C}$ and all morphisms $\alpha ,{\alpha }^{\prime }\in {\text{Hom}}_{\text{C}}(B,Z)$,

$$\alpha \circ f={\alpha }^{\prime }\circ f⟹\alpha ={\alpha }^{\prime }.$$

In other words, $f$ is an epimorphism if it is right cancellable.

In $\text{Set}$, the monomorphisms are precisely the injective functions, and the epimorphisms are precisely the surjective functions.

Several things that we take for granted in $\text{Set}$ break in general categories:

epic + monic $⟹/$ isomorphism

In $\text{Set}$, a function is an isomorphism iff it is both injective and surjective, i.e iff it is both a monomorphism and an epimorphism. But in the category defined by $\le$ on $\mathbb{Z}$, every morphism is both a monomorphism and an epimorphism (since there is at most one morphism between any two objects, the defining conditions become vacuously true), while the only isomorphisms are identities.

epic $⟹/$ right invertible, monic $⟹/$ left invertible

While

$$\begin{array}{rl}& f\text{ is right invertible }⟹f\text{ is an epimorphism},\text{and}\\ & f\text{ is left invertible }⟹f\text{ is a monomorphism}\end{array}$$

can be easily proven universally, the converse is not generally true (It is true, of course, in $\text{Set}$). For example, the map $n\mapsto 2n$ defines a left-cancellable group homomorphism $f:\mathbb{Z}/2\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$. However, there is no group homomorphism $g:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$ such that $gf={\mathrm{id}}_{\mathbb{Z}/2\mathbb{Z}}$. Similarly, the map $n\mapsto n\mathrm{mod}2$ defines a right-cancellable group homomorphism $f:\mathbb{Z}/4\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$. However, since every homomorphism $g:\mathbb{Z}/2\mathbb{Z}\to \mathbb{Z}/4\mathbb{Z}$ must map $[1{]}_2$ to $[2{]}_4$, $f$ is not right invertible.

epic $⟹/$ surjective

In $\text{Set}$, $\text{Gp}$(!) and $\text{Ab}$, epic $⟺$ surjective.

$\text{Ring}$ does not conform to the standards of its brethren, unfortunately. Consider the inclusion homomorphism $\iota :\mathbb{Z}\to \mathbb{Q}$. $\iota$ is an epimorphism in $\text{Ring}$ since if homomorphisms ${\alpha }_1,{\alpha }_2:\mathbb{Q}\to R$ agree on $\mathbb{Z}$, then they must agree on $\mathbb{Q}$:

$${\alpha }_i\left(\frac{p}{q}\right)={\alpha }_i(p){\alpha }_i(q^{-1})={\alpha }_i(p){\alpha }_i(q{)}^{-1}.$$

This makes for another example of a morphism which is monic and epic but not an isomorphism.

monic $⟹/$ injective

We’ll get back to this in a bit.

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Universal Properties

Definition 318.14(Definition).

Let $\text{C}$ be a category.

• We say that $I\in \text{Obj}(\text{C})$ is initial in $\text{C}$ if for all $A\in \text{Obj}(C)$, ${\text{Hom}}_{\text{C}}(I,A)$ is a singleton.
• We say that $F\in \text{Obj}(\text{C})$ is final in $\text{C}$ if for all $A\in \text{Obj}(\text{C})$, ${\text{Hom}}_{\text{C}}(A,F)$ is a singleton.

Example 318.15(Example).

$\mathbb{Z}$ is initial in $\text{Ring}$. Singletons are final in $\text{Set}$.

A category need not have initial and final objects, and when they exist, they may not be unique. However, they are unique up to a unique isomorphism. Initial and final objects are collectively referred to as terminal objects.

Proposition 318.16(Proposition).

Let $\text{C}$ be a category.

• If $I_1,I_2$ are both initial objects in $\text{C}$, then $I_1\cong I_2$.
• If $F_1,F_2$ are both final objects in $\text{C}$, then $F_1\cong F_2$.

Moreover, these isomorphisms are uniquely determined.

Proof.

Since$I_1,I_2$ are initial objects, $|{\text{Hom}}_{\text{C}}(I_1,I_2)|=|{\text{Hom}}_{\text{C}}(I_2,I_1)|=$ $|{\text{Hom}}_{\text{C}}(I_1,I_1)|=|{\text{Hom}}_{\text{C}}(I_2,I_2)|=1$. Let ${\phi }_1\in {\text{Hom}}_{\text{C}}(I_1,I_2)$, ${\phi }_2\in {\text{Hom}}_{\text{C}}(I_2,I_1)$. It follows that ${\phi }_1{\phi }_2={\text{Id}}_{I_2}$ and ${\phi }_2{\phi }_1={\text{Id}}_{{\phi }_1}$, whence ${\phi }_1$ and ${\phi }_2$ are isomorphisms. The same proof works for final objects.□

The same object can be both initial and final, as singletons are in the category of pointed sets.

Definition 318.17(Definition).

We say that a construction satisfies a universal property when it may be viewed as a terminal object of a category.

Example 318.18(Quotienting by equivalence relations).

Let $\sim$ be an equivalence relation defined on a set $A$. Let $\text{C}$ be a category with objects $A\stackrel{\phi }{\to }Z$, where $Z$ is any set, satisfying the property

$$a\sim a^{\prime }⟹\phi (a)=\phi (a^{\prime }).$$

Let objects be denoted by $(\phi ,Z)$. Morphisms $({\phi }_1,Z_1)\to ({\phi }_2,Z_2)$ are commutative diagrams

Denote by $\pi$ the canonical projection from $A$ to $A/\sim$. Then, $(\pi ,A/\sim )$ is an initial object of $\text{C}$. Indeed, for any arbitrary $(\phi ,Z)$ in $\text{C}$, we can find a unique $\overline{\phi }$ such that

commutes.

This information can be sloppily summarized like so:

The quotient $A/\sim$ is universal with respect to the property of mapping $A$ to a set in such a way that equivalent elements have the same image.

Products

Definition 318.19(Universal property of products).

The product of objects $A,B$ in a category $\text{C}$ is the isomorphism class of final objects in the category ${\text{C}}_{A,B}$, as defined in Example 7.

Example 318.20(Products of sets).

Let $\text{C}=\text{Set}$, and let $A,B\in \text{Obj}(\text{C})$. Consider the product $A\times B$ with the two natural projections:

Then for every $(Z,f,g)\in \text{Obj}({\text{C}}_{A,B})$, there exists a unique morphism $\sigma :Z\to A\times B$ such that

commutes. In other words, $\text{Hom}((Z,f,g),(A\times B,{\pi }_A,{\pi }_B))$ is a singleton, whence $(A\times B,{\pi }_A,{\pi }_B)$ is final in ${\text{C}}_{A,B}$.

Exercise 318.21(Products of groups).

As another trivial example, show that for $G,H\in \text{Gp}$, the product group $G\times H$ that we know and love satisfies the universal property of products.

Example 318.22(Product topology).

Coproducts

Definition 318.23(Universal property of coproducts).

The coproduct of objects $A,B$ in a category $\text{C}$ is the isomorphism class of initial objects in the category ${\text{C}}^{A,B}$.

Example 318.24(Disjoint union of Sets).

Let $\text{C}=\text{Set}$. Let $A,B\in \text{Obj}(\text{C})$. Consider the disjoint union $A⨿B$ with the inclusion maps $i_A$ and $i_B$:

Then for every $(Z,f_A,f_B)$, there exists a unique morphism $\sigma :A⨿B\to Z$ such that

commutes. So, $(A⨿B,i_A,i_B)$ is initial in ${\text{C}}^{A,B}$.

Example 318.25(Example).

If $G$ and $H$ are abelian groups, then the product $G\times H$ satisfies the universal property for coproducts in $\text{Ab}$.

Proof.

We need to show that$(G\times H,i_G,i_H)$ is initial in ${\text{C}}^{G,H}$. Let $(A,{\phi }_G,{\phi }_H)\in {\text{C}}^{G,H}$. We need a unique homomorphism $\phi$ which makes the following diagram commute:

We are forced to define $\phi (g,e_H)\equiv {\phi }_G(g)$ and $\phi (e_G,h)\equiv {\phi }_H(h)$ for all $g\in G$ and $h\in H$. Since we require $\phi (g,h)=\phi (g,e_H)\phi (e_G,h)$, our definitions determine $\phi$ over its entire domain. So, we have exactly one candidate for $\phi$. It it a homomorphism?

$$\begin{array}{rl}\phi (a,b)\phi (c,d)& ={\phi }_G(a){\phi }_H(b){\phi }_G(c){\phi }_H(d)\\ & \stackrel{!}{=}{\phi }_G(a){\phi }_G(c){\phi }_H(b){\phi }_H(d)\\ & ={\phi }_G(ac){\phi }_H(bd)\\ & =\phi (ac,bd)\\ & =\phi ((a,b)(c,d)).\end{array}$$

The marked equality is true since $A$ is abelian.□

What are coproducts in $\text{Gp}$?

Exercise 318.26(Exercise).

Prove that the free group $F(\{x,y\})$ is a coproduct $\mathbb{Z}\times \mathbb{Z}$ of $\mathbb{Z}$ by itself in the category $\text{Gp}$.

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Free objects

Definition 318.27(Free objects).

Let $\text{C}$ be a concrete category. Let $X$ be a set. A free object on $X$ is a pair consisting of an object $A$ in $\text{C}$ and an injection $i:X\to A$, that satisfies the following universal property:

For any object $B$ in $\text{C}$ and any map between sets $g:X\to B$, there exists a unique morphism $f:A\to B$ in $\text{C}$ making this diagram commute:

Exercise 318.28(Exercise).

Show that the existence of a free object on the one element set in a concrete category implies that monomorphisms are injective.

Example 318.29(A monic morphism which is not injective).

An abelian group $(G,+)$ is divisible if, for every positive integer $n$ and every $g\in G$, there exists $y\in G$ such that $ny=g$.

In the category of divisible (abelian) groups, the map $\pi :\mathbb{Q}\to \mathbb{Q}/\mathbb{Z}$ is a non-injective monomorphism.

Indeed, suppose that $f,g:G\to \mathbb{Q}$ are such that $\pi \circ f=\pi \circ g$. Let $x\in G$. Then $f(x)-g(x)=n\in \mathbb{Z}$. If $n\ne 0$, then let $y\in G$ be such that $2ny=x$. Then $f(x)=f(2ny)=2nf(y)$, hence $f(y)=\frac{1}{2n}f(x)$; and similarly $g(y)=\frac{1}{2n}g(x)$. Now, $f(y)-g(y)$ must be an integer, but

$$f(y)-g(y)=\frac{1}{2n}(f(x)-g(x))=\frac{1}{2},$$

a contradiction. Therefore, $n=0$, so $f(x)=g(x)$. Thus, $f=g$ and $\pi$ is a monomorphism.