## 4.2 Definitions

We recall the definitions, partly to fix notation.

Definition 4.2.1. A *category* $\mathcal{C}$ consists of the following data:

A set of objects $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

For each pair $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a set of morphisms $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$.

For each triple $x, y, z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a composition map $ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, z) \times \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, z) $, denoted $(\phi , \psi ) \mapsto \phi \circ \psi $.

These data are to satisfy the following rules:

For every element $x\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a morphism $\text{id}_ x\in \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, x)$ such that $\text{id}_ x \circ \phi = \phi $ and $\psi \circ \text{id}_ x = \psi $ whenever these compositions make sense.

Composition is associative, i.e., $(\phi \circ \psi ) \circ \chi = \phi \circ ( \psi \circ \chi )$ whenever these compositions make sense.

It is customary to require all the morphism sets $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ to be disjoint. In this way a morphism $\phi : x \to y$ has a unique *source* $x$ and a unique *target* $y$. This is not strictly necessary, although care has to be taken in formulating condition (2) above if it is not the case. It is convenient and we will often assume this is the case. In this case we say that $\phi $ and $\psi $ are *composable* if the source of $\phi $ is equal to the target of $\psi $, in which case $\phi \circ \psi $ is defined. An equivalent definition would be to define a category as a quintuple $(\text{Ob}, \text{Arrows}, s, t, \circ )$ consisting of a set of objects, a set of morphisms (arrows), source, target and composition subject to a long list of axioms. We will occasionally use this point of view.

Definition 4.2.4. A morphism $\phi : x \to y$ is an *isomorphism* of the category $\mathcal{C}$ if there exists a morphism $\psi : y \to x$ such that $\phi \circ \psi = \text{id}_ y$ and $\psi \circ \phi = \text{id}_ x$.

An isomorphism $\phi $ is also sometimes called an *invertible* morphism, and the morphism $\psi $ of the definition is called the *inverse* and denoted $\phi ^{-1}$. It is unique if it exists. Note that given an object $x$ of a category $\mathcal{A}$ the set of invertible elements $\text{Aut}_\mathcal {A}(x)$ of $\mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(x, x)$ forms a group under composition. This group is called the *automorphism* group of $x$ in $\mathcal{A}$.

Definition 4.2.5. A *groupoid* is a category where every morphism is an isomorphism.

Example 4.2.6. A group $G$ gives rise to a groupoid with a single object $x$ and morphisms $\mathop{\mathrm{Mor}}\nolimits (x, x) = G$, with the composition rule given by the group law in $G$. Every groupoid with a single object is of this form.

Example 4.2.7. A set $C$ gives rise to a groupoid $\mathcal{C}$ defined as follows: As objects we take $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C}) := C$ and for morphisms we take $\mathop{\mathrm{Mor}}\nolimits (x, y)$ empty if $x\neq y$ and equal to $\{ \text{id}_ x\} $ if $x = y$.

Definition 4.2.8. A *functor* $F : \mathcal{A} \to \mathcal{B}$ between two categories $\mathcal{A}, \mathcal{B}$ is given by the following data:

A map $F : \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \to \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$.

For every $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ a map $F : \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(x, y) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {B}(F(x), F(y))$, denoted $\phi \mapsto F(\phi )$.

These data should be compatible with composition and identity morphisms in the following manner: $F(\phi \circ \psi ) = F(\phi ) \circ F(\psi )$ for a composable pair $(\phi , \psi )$ of morphisms of $\mathcal{A}$ and $F(\text{id}_ x) = \text{id}_{F(x)}$.

Note that every category $\mathcal{A}$ has an *identity* functor $\text{id}_\mathcal {A}$. In addition, given a functor $G : \mathcal{B} \to \mathcal{C}$ and a functor $F : \mathcal{A} \to \mathcal{B}$ there is a *composition* functor $G \circ F : \mathcal{A} \to \mathcal{C}$ defined in an obvious manner.

Definition 4.2.9. Let $F : \mathcal{A} \to \mathcal{B}$ be a functor.

We say $F$ is *faithful* if for any objects $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ the map

\[ F : \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(x, y) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {B}(F(x), F(y)) \]

is injective.

If these maps are all bijective then $F$ is called *fully faithful*.

The functor $F$ is called *essentially surjective* if for any object $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ there exists an object $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ such that $F(x)$ is isomorphic to $y$ in $\mathcal{B}$.

Definition 4.2.10. A *subcategory* of a category $\mathcal{B}$ is a category $\mathcal{A}$ whose objects and arrows form subsets of the objects and arrows of $\mathcal{B}$ and such that source, target and composition in $\mathcal{A}$ agree with those of $\mathcal{B}$ and such that the identity morphism of an object of $\mathcal{A}$ matches the one in $\mathcal{B}$. We say $\mathcal{A}$ is a *full subcategory* of $\mathcal{B}$ if $\mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(x, y) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {B}(x, y)$ for all $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. We say $\mathcal{A}$ is a *strictly full* subcategory of $\mathcal{B}$ if it is a full subcategory and given $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ any object of $\mathcal{B}$ which is isomorphic to $x$ is also in $\mathcal{A}$.

If $\mathcal{A} \subset \mathcal{B}$ is a subcategory then the identity map is a functor from $\mathcal{A}$ to $\mathcal{B}$. Furthermore a subcategory $\mathcal{A} \subset \mathcal{B}$ is full if and only if the inclusion functor is fully faithful. Note that given a category $\mathcal{B}$ the set of full subcategories of $\mathcal{B}$ is the same as the set of subsets of $\mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$.

Example 4.2.12. A homomorphism $p : G\to H$ of groups gives rise to a functor between the associated groupoids in Example 4.2.6. It is faithful (resp. fully faithful) if and only if $p$ is injective (resp. an isomorphism).

Example 4.2.13. Given a category $\mathcal{C}$ and an object $X\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we define the *category of objects over $X$*, denoted $\mathcal{C}/X$ as follows. The objects of $\mathcal{C}/X$ are morphisms $Y\to X$ for some $Y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Morphisms between objects $Y\to X$ and $Y'\to X$ are morphisms $Y\to Y'$ in $\mathcal{C}$ that make the obvious diagram commute. Note that there is a functor $p_ X : \mathcal{C}/X\to \mathcal{C}$ which simply forgets the morphism. Moreover given a morphism $f : X'\to X$ in $\mathcal{C}$ there is an induced functor $F : \mathcal{C}/X' \to \mathcal{C}/X$ obtained by composition with $f$, and $p_ X\circ F = p_{X'}$.

Example 4.2.14. Given a category $\mathcal{C}$ and an object $X\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we define the *category of objects under $X$*, denoted $X/\mathcal{C}$ as follows. The objects of $X/\mathcal{C}$ are morphisms $X\to Y$ for some $Y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Morphisms between objects $X\to Y$ and $X\to Y'$ are morphisms $Y\to Y'$ in $\mathcal{C}$ that make the obvious diagram commute. Note that there is a functor $p_ X : X/\mathcal{C}\to \mathcal{C}$ which simply forgets the morphism. Moreover given a morphism $f : X'\to X$ in $\mathcal{C}$ there is an induced functor $F : X/\mathcal{C} \to X'/\mathcal{C}$ obtained by composition with $f$, and $p_{X'}\circ F = p_ X$.

Definition 4.2.15. Let $F, G : \mathcal{A} \to \mathcal{B}$ be functors. A *natural transformation*, or a *morphism of functors* $t : F \to G$, is a collection $\{ t_ x\} _{x\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})}$ such that

$t_ x : F(x) \to G(x)$ is a morphism in the category $\mathcal{B}$, and

for every morphism $\phi : x \to y$ of $\mathcal{A}$ the following diagram is commutative

\[ \xymatrix{ F(x) \ar[r]^{t_ x} \ar[d]_{F(\phi )} & G(x) \ar[d]^{G(\phi )} \\ F(y) \ar[r]^{t_ y} & G(y) } \]

Sometimes we use the diagram

\[ \xymatrix{ \mathcal{A} \rtwocell ^ F_ G{t} & \mathcal{B} } \]

to indicate that $t$ is a morphism from $F$ to $G$.

Note that every functor $F$ comes with the *identity* transformation $\text{id}_ F : F \to F$. In addition, given a morphism of functors $t : F \to G$ and a morphism of functors $s : E \to F$ then the *composition* $t \circ s$ is defined by the rule

\[ (t \circ s)_ x = t_ x \circ s_ x : E(x) \to G(x) \]

for $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$. It is easy to verify that this is indeed a morphism of functors from $E$ to $G$. In this way, given categories $\mathcal{A}$ and $\mathcal{B}$ we obtain a new category, namely the category of functors between $\mathcal{A}$ and $\mathcal{B}$.

Definition 4.2.17. An *equivalence of categories* $F : \mathcal{A} \to \mathcal{B}$ is a functor such that there exists a functor $G : \mathcal{B} \to \mathcal{A}$ such that the compositions $F \circ G$ and $G \circ F$ are isomorphic to the identity functors $\text{id}_\mathcal {B}$, respectively $\text{id}_\mathcal {A}$. In this case we say that $G$ is a *quasi-inverse* to $F$.

Lemma 4.2.18. Let $F : \mathcal{A} \to \mathcal{B}$ be a fully faithful functor. Suppose for every $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ we are given an object $j(X)$ of $\mathcal{A}$ and an isomorphism $i_ X : X \to F(j(X))$. Then there is a unique functor $j : \mathcal{B} \to \mathcal{A}$ such that $j$ extends the rule on objects, and the isomorphisms $i_ X$ define an isomorphism of functors $\text{id}_\mathcal {B} \to F \circ j$. Moreover, $j$ and $F$ are quasi-inverse equivalences of categories.

**Proof.**
To construct $j : \mathcal{B} \to \mathcal{A}$, there are two steps. Firstly, we define the map $j : \mathop{\mathrm{Ob}}\nolimits (\mathcal{B}) \to \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ that associates $j(X)$ to $X \in \mathcal{B}$. Secondly, if $X,Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ and $\phi : X \to Y$, we consider $\phi ' := i_ Y \circ \phi \circ i_ X^{-1}$. There is an unique $\varphi $ verifying $F(\varphi ) = \phi '$, using that $F$ is fully faithful. We define $j(\phi ) = \varphi $. We omit the verification that $j$ is a functor. By construction the diagram

\[ \xymatrix{ X \ar[r]_-{i_ X} \ar[d]_{\phi } & F(j(X)) \ar[d]^{F\circ j(\phi )} \\ Y \ar[r]^-{i_ Y} & F(j(Y)) } \]

commutes. Hence, as each $i_ X$ is an isomorphism, $\{ i_ X\} _ X$ is an isomorphism of functors $\text{id}_\mathcal {B} \to F\circ j$. To conclude, we have to also prove that $j \circ F$ is isomorphic to $\text{id}_\mathcal {A}$. However, since $F$ is fully faithful, in order to do this it suffices to prove this after post-composing with $F$, i.e., it suffices to show that $F \circ j \circ F$ is isomorphic to $F \circ \text{id}_\mathcal {A}$ (small detail omitted). Since $F \circ j \cong \text{id}_\mathcal {B}$ this is clear.
$\square$

Lemma 4.2.19. A functor is an equivalence of categories if and only if it is both fully faithful and essentially surjective.

**Proof.**
Let $F : \mathcal{A} \to \mathcal{B}$ be essentially surjective and fully faithful. As by convention all categories are small and as $F$ is essentially surjective we can, using the axiom of choice, choose for every $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ an object $j(X)$ of $\mathcal{A}$ and an isomorphism $i_ X : X \to F(j(X))$. Then we apply Lemma 4.2.18 using that $F$ is fully faithful.
$\square$

Definition 4.2.20. Let $\mathcal{A}$, $\mathcal{B}$ be categories. We define the *product category* $\mathcal{A} \times \mathcal{B}$ to be the category with objects $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A} \times \mathcal{B}) = \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \times \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ and

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{A} \times \mathcal{B}}((x, y), (x', y')) := \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(x, x')\times \mathop{\mathrm{Mor}}\nolimits _\mathcal {B}(y, y'). \]

Composition is defined componentwise.

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