Definitions and Examples - Category Theory

1 Definitions and Examples

Definition 1.1 (Category). A category $C$ consists of:

(a)
a collection $ob C$ of objects $A, B, C, \dots$ .
(b)
a collection $mor C$ of morphisms $f, g, h, \dots$ .
(c)
two operations $dom$ , $cod$ from $mor C$ to $ob C$ : we write $f : A \to B$ for “ $f$ is a morphism and $dom f = A$ and $cod f = B$ ”.
(d)
an operation from $ob C$ to $mor C$ sending $A$ to $1_{A} : A \to A$ .
(e)
a partial binary operation $(f, g) \mapsto f g$ on $mor C$ , such that $f g$ is defined if and only if $dom f = cod g$ , and in this case we have $dom f g = dom g$ and $cod f g = cod f$ .

These are subject to the axioms:

(f)
$f 1_{A} = f$ and $1_{A} g = g$ when the composites are defined.
(g)
$f (g h) = (f g) h$ whenever $f g$ and $g h$ are defined.

Remark 1.2.

(a) $ob C$ and $mor C$ needn’t be sets. If they are, we call $C$ a small category.
(b) We could formalise the definition without mentioning objects, but we don’t.
(c) $f g$ means “first $g$ , then $f$ ”.

Example 1.3.

(a) $S e t =$ category of all sets and the functions between them. (Formally, a morphism of $S e t$ is a pair $(f, B)$ where $f$ is a set-theoretic function, and $B$ is its codomain.)
(b) We have categories:
- $G r o u p$ of groups and group homomorphisms
- $R n g$ of rings and homomorphisms
- ${V e c t}_{k}$ of vector spaces over a field $k$
- and so on
(c) We have categories
- $T o p$ of topological spaces and continuous maps
- $M e t$ of metric spaces and non-expansive maps (i.e. $f$ such that $d (f (x), f (y)) \leq d (x, y)$ )
- $M f d$ of smooth manifolds and $C^{\infty}$ maps
Also $T o p G p$ for topological groups and continuous homomorphisms, etc...
(d) We have a category $H t p y$ with the same objects as $T o p$ , but morphisms $X \to Y$ are homotopy classes of continuous maps.
In general, given $C$ and an equivalence relation $\equiv$ on $mor C$ such that
$f \equiv g ⟹ dom f = dom g and cod f = cod g$

and
$f \equiv g ⟹ f g \equiv g h and k f \equiv k g when the composites are defined$

we can form a quotient category $C ∕ \equiv$ .
(e) The category $R e l$ has the same objects as $S e t$ , but morphisms $A \to B$ are relations $R \subseteq A \times B$ , with composition defined by
$R \circ S = {(a, c) | (\exists b) (a, b) \in S \land (b, c) \in R} .$

We can also define the category $P a r t$ of sets with partial functions.
(f) For any category $C$ , the opposite category $C^{o p}$ has the same objects and morphisms as $C$ but $dom$ and $cod$ are interchanged and composition is reversed.
This yields a duality principle: if $P$ is a true statement about categories, so is $P^{*}$ obtained by reversing arrows in $P$ .
(g) A (small) category with one object $*$ is a monoid (a semigroup with an identity). In particular, a group is a $1 -$ object small category whose morphisms are all isomorphisms.
(h) A groupoid is a category whose morphisms are all isomorphisms. For example, the fundamental groupoid $π_{1} (X)$ of a topological space $X$ has points of $X$ as objects, and morphisms $x \to y$ are homotopy classes of paths from $x$ to $y$ (c.f. the fundamental group $π_{1} (X, x)$ ).
(i) A discrete category is one whose only morphisms are identities. If $C$ is such that for any pair of objects $(A, B)$ there is at most one morphism $A \to B$ then $mor C$ becomes a reflexive, transitive relation on $ob C$ . We call such a $C$ a preorder. In particular, a poset is a small preorder whose only isomorphisms are identities.
(j) Given a field $k$ , the category ${M a t}_{k}$ has natural numbers as objects, and morphisms $n \to p$ are $p \times n$ matrices, with entries from $k$ , and composition is matrix multiplication.

Definition 1.4 (Functor). Let $C$ and $D$ be categories. A functor $F : C \to D$ consists of mappings $F : ob C \to ob D$ and $F + mor C \to mor D$ such that:

$F (dom f) = dom F f$
$F (cod f) = cod F f$
$F (1_{A}) = 1_{F A}$
$F (f g) = (F f) (F g)$ whenever $f g$ is defined.

We write $Cat$ for the category of small categories and the functors between them.

Example 1.5.

(a) We have forgetful functors $G p \to S e t$ , $R n g \to S e t$ , $T o p \to S e t$ , … or slightly more interestingly, $R n g \to A b G p$ , $M e t \to T o p$ , $T o p G p \to T o p$ , $T o p G p \to G p$ , …
(b) The construction of free groups is a functor $S e t \to G p$ : given a set $A$ , $F A$ is the group freely generated by $A$ , such that every mapping $A \to G$ where $G$ has a group structure extends uniquely to a homomorphism $F A \to G$ . Given $A \overset{f}{\to} B$ , we define $F f : F A \to F B$ to be the unique homomorphism extending $A \overset{f}{\to} B ↪ F B$ . If we also have $B \overset{g}{\to} C$ , $F (g f)$ and $(F g) (F f)$ are both homomorphisms extending $A \overset{f}{\to} B \overset{g}{\to} C ↪ F C$ .
(c) Given a set $A$ , we define $P A$ to be the set of subsets of $A$ . Given $f : A \to B$ , we define $P f : P A \to P B$ by $P f (A^{'}) = f (a) | a \in A^{'} \subseteq B$ . So $P$ is a functor $S e t \to S e t$ .
(d) But we also have a functor $P^{*} : {S e t}^{o p} \to S e t$ (or $S e t \to {S e t}^{o p}$ ): $P^{*} A = P A$ and, for $A \overset{f}{\to} B$ , $G^{*} f : P B \to P A$ is given by $P^{*} f (B^{'}) = a i n A | f (a) \in B^{'}$ . We use the term “contravariant functor $C \to D$ ” for a functor $C \to D^{o p}$ .
(e) Given a vector space $V$ over $k$ , we write $V^{*}$ for the space of linear maps $V \to k$ . Given $f : V \to W$ , we write $f^{*} : W^{*} \to V^{*}$ for the mapping $𝜃 \mapsto 𝜃 f$ . This defines a functor ${(∙)}^{*} : {V e c t}_{k}^{o p} \to {V e c t}_{k}$ .
(f) The mapping $C \mapsto C^{o p}$ , $F \mapsto F$ defines a functor $C a t \to C a t$ .
(g) A functor between monoids is a monoid homomorphism; a functor between posets is a monotone map.
(h) Given a group $G$ , a functor $G \to S e t$ is given by a set $A$ equipped with a $G$ -action $(g, a) \mapsto g \cdot a$ , i.e. a permutation representation of $G$ . Similarly, a functor $G \to {V e c t}_{k}$ is a $k$ -linear representation of $G$ .
(i) The fundamental group construction is a functor $Π_{1} : {T o p}_{*} \to G p$ , where ${T o p}_{*}$ is the category of topological spaces with basepoints, and morphisms being the continuous maps which preserve the basepoints.

Definition 1.6 (Natural transformation). Given categories $C$ and $D$ , and two functors $C ⇉_{G}^{F} D$ , a natural transformation $α : F \to G$ assigns to each $A \in ob C$ a morphism $α_{A} : F A \to G A$ in $D$ , such that for any $A \overset{f}{\to} B$ in $C$ , the square

commutes (we call this square the naturality square for

α

f

). Given

α

as above, and

β : G \to H

, we define

β α : F \to H

{(β α)}_{A} = β_{A} α_{A}

. We write

[C, D]

for the category of functors

C \to D

and natural transformations between them.

Example 1.7.

(a) Given a vector space $V$ , we have a linear map $α_{V} : V \to V^{* *}$ sending $v \in V$ to the linear form $𝜃 \mapsto 𝜃 (v)$ on $V^{* *}$ . These maps define a natural transformation $1_{{V e c t}_{k}} \to {(∙)}^{* *}$ .
(b) There is a natural transformation $α : 1_{S e t} \to U F$ , where $F$ is the free group functor and $U$ is the forgetful functor $G p \to S e t$ , whose value at $A$ is the inclusion $A ↪ U F A$ . The naturality square
commutes by the definition of $F f$ .
(c) For any $A$ , we have a mapping $η_{A} : A \to P A$ given by $A η_{A} (a) = {a}$ . This is a natural transformation $1_{S e t} \to P$ since $P f ({a}) = {f (a)}$ for any $a \in A$ .
(d) Given order-preserving maps $P ⇉_{g}^{f} Q$ between posets, there exists a unique natural transformation $f \to g$ if and only if $f (p) \leq g (p)$ for all $P \in P$ .
(e) Given two group homomorphisms $G ⇉_{v}^{u} H$ , a natural transformation $u \to v$ is given by $h \in H$ such that $h u (g) = v (g) h$ for all $g \in G$ , or equivalently $u (g) = h^{- 1} v (g) h$ , i.e. $u$ and $v$ are conjugate homomorphisms. In particular, the group of natural transformations $u \to u$ is the centraliser of the image of $u$ .
(f) If $A$ and $B$ are $G$ -sets considered as functors $G \to S e t$ , a natural transformation $f : A \to B$ is a $G$ -invariant map, i.e. $f : A \to B$ such that $g f (a) = f (g a)$ for all $a \in A$ , $g \in G$ .
(g) The Hurewicz homomorphism links the homotopy and homology groups of a space $X$ . Elements of $π_{n} (X, x)$ are homotopy classes of basepoint-preserving maps $S^{n} \overset{f}{\to} X$ . If we think of $S^{n}$ as $\partial Δ^{n + 1}$ , $f$ defines a singular $n$ -cycle on $X$ and homotopic maps differ by an $n$ -boundary, so we get a well-defined map $π_{n} (X, x) \overset{h_{n}}{\to} H_{n} (X)$ . $h_{n}$ is a homomorphism, and it’s a natural transformation $π_{n} \to H_{n} U$ , where $U$ is the forgetful functor ${T o p}_{*} \to T o p$ .

We have isomorphisms of categories: e.g. $F : R e l \to {R e l}^{o p}$ defined by $F A = A$ , $F R = R^{o} = {(b, a) | (a, b) \in R}$ is its own inverse.

But we have a weaker notion of equivalence of categories.

Lemma 1.8. Assuming that:

$α : F \to G$ is a natural transformation between functors $C ⇉_{}^{} D$

Then

α

is an isomorphism in

[C, D]

if and only if

α_{A}

is an isomorphism in

D

for each

A

Proof.

$\Rightarrow$ Obvious since composition in $[C, D]$ .
$\Leftarrow$ Suppose each $α_{A}$ has an inverse $β_{A}$ . Given $A \overset{f}{\to} B$ in $C$ , in the diagram
we have $β_{B} (G f) = β_{B} (G f) α_{A} β_{A} = β_{B} α_{B} (F f) β_{A} = (F f) β_{A}$ . □

Definition 1.9 (Equivalence of categories). Let $C$ and $D$ be categories. An equivalence between $C$ and $D$ consists of functors $F : C \to D$ and $G : D \to C$ together with natural isomorphisms $α : 1_{C} \to G F$ , $β : F G \to 1_{D}$ . We write $C \equiv D$ if there exists an equivalence between $C$ and $D$ .

We say $P$ is a categorical property if

(C has P and C \equiv D) ⟹ D has P .

Example 1.10.

(a) The category $P a r t$ of sets and partial functions is equivalent to ${S e t}_{*}$ (the category of pointed sets). We define $F : {S e t}_{*} \to P a r t$ by $F (A, a) = A ∖ {a}$ and if $f : (A, a) \to (B, b)$ , with $(F f) (x) = f (x)$ if $f (x) \neq b$ and undefined otherwise. Then define $G : P a r t \to {S e t}_{*}$ by $G (A) = (A \cup {A}, A)$ and if $f : A ⇁ B$ , then
$G f (x) = {\begin{matrix} f (x) & if x \in A and f (x) is defined \\ B & otherwise \end{matrix} .$

Then $F G = 1_{P a r t}$ ; $G F \neq 1_{{S e t}_{*}}$ , but there is an isomorphism $1_{{S e t}_{*}} \to G F$ . Note that $P a r t ⁄ ≅ {S e t}_{*}$ .
(b) We have an equivalence ${f d V e c t}_{k} \equiv {f d V e c t}_{k}^{o p}$ : both functors are ${(∙)}^{*}$ , and both isomorphisms are $α : 1_{{f d V e c t}_{k}} \to {(∙)}^{* *}$ .
(c) We have an equivalence ${f d V e c t}_{k} \equiv {M a t}_{k}$ : we define $F : {M a t}_{k} \to {f d V e c t}_{k}$ by $F (n) = k^{n}$ , $F (n \overset{A}{\to} p)$ is the linear map $k^{n} \to k^{p}$ represented by $A$ (with respect to standard bases). TO define $G$ , choose a basis for each $V$ , and define $G (V) = \dim V$ ,
$G (V \overset{f}{\to} W) = matrix representing f with respect to chosen bases .$

$G F = 1_{{M a t}_{k}}$ ; the choice of bases yields isomorphisms $k^{\dim V} \to V$ for each $V$ , which form a natural transformation $F G \to 1_{{f d V e c t}_{k}}$ .

Definition 1.11 (Faithful / full / essentially surjective). Let $F : C \to D$ be a functor.

(a)
We say $F$ is faithful if, given $f$ and $g$ in $mor C$ , $(F f = F g$ , $dom f = dom g$ , $cod f = cod g) ⟹ f = g$ .
(b)
We say $F$ is full if, for every $g : F A \to F B$ in $D$ , there exists $f : A \to B$ in $C$ with $F f = g$ .
(c)
We say $F$ is essentially surjective if, for any $B \in ob D$ , there exists $A \in ob C$ with $F A ≅ B$ .

Note that if $F$ is full and faithful, it’s essentially injective: given $F A \to_{≅}^{g} F B$ in $D$ , the unique $A \overset{f}{\to} B$ with $F f = g$ is an isomorphism.

We say $D \subseteq C$ is a full subcategory if the inclusion $D \to C$ is a full functor.

Lemma 1.12. Assuming that:

$F : C \to D$

Then

F

is part of an equivalence

C \equiv D

if and only if

F

is full, faithful, essentially surjective.

Proof.

$\Rightarrow$ Suppose give $G$ , $α$ and $β$ as in Definition 1.9. Then $β_{B} : F G B \to B$ witnesses the fact that $F$ is essentially surjective. If $A ⇉_{g}^{f} B$ satisfy $F g = F g$ , then $G F f = G F g$ ; but $f = α_{B}^{- 1} (G F f) α_{A}$ , so $f = g$ . Suppose given $F A \overset{g}{\to} F B$ ; then $f = α_{B}^{- 1} (G g) α_{A}$ satisfies $G F f = G f$ but $G$ is faithful for the same reason as $F$ , so $F f = g$ .
$\Leftarrow$ For each $B \in ob D$ , chose $G B \in ob C$ and an isomorphism $β_{B} : F G B \to B$ . Given $B \overset{g}{\to} C$ , define $G g : G B \to G C$ to be the unique morphism such that $F G g = β_{C}^{- 1} g β_{B}$ . Functoriality follows from uniqueness, and naturality of $β$ . We define $α_{A} : A \to G F A$ to be the unique morphism such that $F α_{A} = β_{F A}^{- 1} : F A \to F G F A$ . $α_{A}$ is an isomorphism, and naturality squares for $α$ are mapped by $F$ to naturality squares for $β^{- 1}$ , so they commute. □

Definition 1.13 (Skeleton). By a skeleton of a category $C$ , we mean a full subcategory containing just one object from each isomorphism class.

We say $C$ is skeletal if it’s a skeleton of itself.

Example. ${M a t}_{k}$ is a skeletal category; it’s isomorphic to the skeleton of ${f d V e c t}_{k}$ consisting of the spaces $k^{n}$ .

However, working with skeletal categories involves heavy use of the axiom of choice.

Definition 1.14 (Monomorphism / epimorphism). Let $f : A \to B$ be a morphism in a category $C$ . We say $f$ is a monomorphism (or monic) if, given $C ⇉_{h}^{g} A$ , $f g = f h ⟹ g = h$ . We say $f$ is an epimorphism (or epic) if it’s a monomorphism in $C^{o p}$ .

We write $A \overset{f}{↣} B$ to indicate that $f$ is monic, and $A \overset{f}{↠} B$ to indicate that it’s epic.

We say $C$ is balanced if every arrow which is monic and epic is an isomorphism.

We will call a monic morphism $e$ split if it has a left inverse (and similarly we may define the notion of split epic).

Example 1.15.

(a) In $S e t$ , monic $⟺$ injective ( $\Leftarrow$ obvious; for $\Rightarrow$ consider morphisms ${*} \to A$ ). Also, epic $⟺$ surjective ( $\Leftarrow$ obvious; for $\Rightarrow$ consider morphisms $B \to {0, 1}$ ).
(b) In $G p$ , monic $⟺$ injective (for $\Rightarrow$ consider homomorphisms $ℤ \to G$ ), and epic $⟺$ surjective (but $\Rightarrow$ is quite non-trivial – it uses free products with amalgamation).
(c) In $R n g$ , monic $⟺$ injective, but epic does not imply surjective (for example, consider $ℤ ↪ ℚ$ ).
(d) In $T o p$ , monic $⟺$ injective and epic $⟺$ surjective (as in $S e t$ ) but $T o p$ isn’t balanced.
(e) In preorder, all morphisms are monic and epic, so a preorder is balanced if and only if it’s an equivalence relation.