Subsection 7.3.1 (00P0): The Graph of a Function

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7.3.1 The Graph of a Function

Let $f\colon A\to B$ be a function.

Definition 7.3.1.1.1 ▶ The Graph of a Function

The graph of $f$ is the relation $\text{Gr}\webleft (f\webright )\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ defined as follows:¹

Viewing relations from $A$ to $B$ as subsets of $A\times B$, we define
\[ \text{Gr}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ \webleft (a,f\webleft (a\webright )\webright )\in A\times B\ \middle |\ a\in A\webright\} ; \]
Viewing relations from $A$ to $B$ as functions $A\times B\to \{ \mathsf{true},\mathsf{false}\} $, we define

\[ \webleft [\text{Gr}\webleft (f\webright )\webright ]\webleft (a,b\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\begin{cases} \mathsf{true}& \text{if $b=f\webleft (a\webright )$,}\\ \mathsf{false}& \text{otherwise} \end{cases} \]

for each $\webleft (a,b\webright )\in A\times B$;
Viewing relations from $A$ to $B$ as functions $A\to \mathcal{P}\webleft (B\webright )$, we define

\[ \webleft [\text{Gr}\webleft (f\webright )\webright ]\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ f\webleft (a\webright )\webright\} \]

for each $a\in A$, i.e. we define $\text{Gr}\webleft (f\webright )$ as the composition

\[ A \overset {f}{\to } B \overset {\chi _{B}}{\to } \mathcal{P}\webleft (B\webright ). \]

¹Further Notation: We write $\text{Gr}\webleft (A\webright )$ for $\text{Gr}\webleft (\text{id}_{A}\webright )$, and call it the graph of $A$.

Proposition 7.3.1.1.2 ▶ Properties of Graphs of Functions

Let $f\colon A\to B$ be a function.

Functoriality. The assignment $A\mapsto \text{Gr}\webleft (A\webright )$ defines a functor
\[ \text{Gr}\colon \mathsf{Sets}\to \mathrm{Rel} \]

where
- Action on Objects. For each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
  
  \[ \text{Gr}\webleft (A\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}A; \]
- Action on Morphisms. For each $A,B\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, the action on $\textup{Hom}$-sets
  
  \[ \text{Gr}_{A,B}\colon \mathsf{Sets}\webleft (A,B\webright ) \to \underbrace{\mathrm{Rel}\webleft (\text{Gr}\webleft (A\webright ),\text{Gr}\webleft (B\webright )\webright )}_{\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{Rel}\webleft (A,B\webright )} \]
  
  of $\text{Gr}$ at $\webleft (A,B\webright )$ is defined by
  
  \[ \text{Gr}_{A,B}\webleft (f\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Gr}\webleft (f\webright ), \]
  
  where $\text{Gr}\webleft (f\webright )$ is the graph of $f$ as in Definition 7.3.1.1.1.
In particular:
- Preservation of Identities. We have
  
  \[ \text{Gr}\webleft (\text{id}_{A}\webright )=\chi _{A} \]
  
  for each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
- Preservation of Composition. We have
  
  \[ \text{Gr}\webleft (g\circ f\webright )=\text{Gr}\webleft (g\webright )\mathbin {\diamond }\text{Gr}\webleft (f\webright ) \]
  
  for each pair of functions $f\colon A\to B$ and $g\colon B\to C$.
Adjointness Inside $\textbf{Rel}$. We have an adjunction
in $\textbf{Rel}$, where $f^{-1}$ is the inverse of $f$ of Definition 7.3.2.1.1.
Adjointness. We have an adjunction
witnessed by a bijection of sets
\[ \mathrm{Rel}\webleft (\text{Gr}\webleft (A\webright ),B\webright ) \cong \mathsf{Sets}\webleft (A,\mathcal{P}\webleft (B\webright )\webright ) \]

natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and $B\in \text{Obj}\webleft (\mathrm{Rel}\webright )$.
Interaction With Inverses. We have
\begin{align*} \text{Gr}\webleft (f\webright )^{\dagger } & = f^{-1},\\ \webleft (f^{-1}\webright )^{\dagger } & = \text{Gr}\webleft (f\webright ). \end{align*}
Cocontinuity. The functor $\text{Gr}\colon \mathsf{Sets}\to \mathrm{Rel}$ of Item 1 preserves colimits.
Characterisations. Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation. The following conditions are equivalent:
1. There exists a function $f\colon A\to B$ such that $R=\text{Gr}\webleft (f\webright )$.
2. The relation $R$ is total and functional.
3. The weak and strong inverse images of $R$ agree, i.e. we have $R^{-1}=R_{-1}$.
4. The relation $R$ has a right adjoint $R^{\dagger }$ in $\mathrm{Rel}$.

Proof of Proposition 7.3.1.1.2.

Item 1: Functoriality

Clear.

Item 2: Adjointness Inside $\textbf{Rel}$

We need to check that there are inclusions

\begin{align*} \chi _{A} & \subset f^{-1}\mathbin {\diamond }\text{Gr}\webleft (f\webright ),\\ \text{Gr}\webleft (f\webright )\mathbin {\diamond }f^{-1} & \subset \chi _{B}. \end{align*}

These correspond respectively to the following conditions:

For each $a\in A$, there exists some $b\in B$ such that $a\sim _{\text{Gr}\webleft (f\webright )}b$ and $b\sim _{f^{-1}}a$.
For each $a,b\in A$, if $a\sim _{\text{Gr}\webleft (f\webright )}b$ and $b\sim _{f^{-1}}a$, then $a=b$.

In other words, the first condition states that the image of any $a\in A$ by $f$ is nonempty, whereas the second condition states that $f$ is not multivalued. As $f$ is a function, both of these statements are true, and we are done.

Item 3: Adjointness

The stated bijection follows from Chapter 6: Relations,

, with naturality being clear.

Item 4: Interaction With Inverses

Clear.

Item 5: Cocontinuity

Omitted.

Item 6: Characterisations

We claim that Item (a), Item (b), Item (c), and Item (d) are indeed equivalent:

Item (a)$\iff $Item (b). This is shown in the proof of of .
Item (b)$\implies $Item (c). If $R$ is total and functional, then, for each $a\in A$, the set $R\webleft (a\webright )$ is a singleton, implying that

\begin{align*} R^{-1}\webleft (V\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ a\in A \ \middle |\ R\webleft (a\webright )\cap V\neq \text{Ø}\webright\} ,\\ R_{-1}\webleft (V\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ a\in A\ \middle |\ R\webleft (a\webright )\subset V\webright\} \end{align*}

are equal for all $V\in \mathcal{P}\webleft (B\webright )$, as the conditions $R\webleft (a\webright )\cap V\neq \text{Ø}$ and $R\webleft (a\webright )\subset V$ are equivalent when $R\webleft (a\webright )$ is a singleton.
Item (c)$\implies $Item (b). We claim that $R$ is indeed total and functional:
- Totality. If we had $R\webleft (a\webright )=\text{Ø}$ for some $a\in A$, then we would have $a\in R_{-1}\webleft (\text{Ø}\webright )$, so that $R_{-1}\webleft (\text{Ø}\webright )\neq \text{Ø}$. But since $R^{-1}\webleft (\text{Ø}\webright )=\text{Ø}$, this would imply $R_{-1}\webleft (\text{Ø}\webright )\neq R^{-1}\webleft (\text{Ø}\webright )$, a contradiction. Thus $R\webleft (a\webright )\neq \text{Ø}$ for all $a\in A$ and $R$ is total.
- Functionality. If $R^{-1}=R_{-1}$, then we have
  
  \begin{align*} \webleft\{ a\webright\} & = R^{-1}\webleft (\webleft\{ b\webright\} \webright )\\ & = R_{-1}\webleft (\webleft\{ b\webright\} \webright ) \end{align*}
  
  for each $b\in R\webleft (a\webright )$ and each $a\in A$, and thus $R\webleft (a\webright )\subset \webleft\{ b\webright\} $. But since $R$ is total, we must have $R\webleft (a\webright )=\webleft\{ b\webright\} $, and thus we see that $R$ is functional.
Item (a)$\iff $Item (d). This follows from Chapter 6: Relations, Proposition 6.3.3.1.1.

This finishes the proof.

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