Subsection 2.3.5 (003A): Sets of Maps

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The set of maps from $A$ to $B$¹ is the set $\mathsf{Sets}\webleft (A,B\webright )$² whose elements are the functions from $A$ to $B$.

¹Further Terminology: Also called the Hom set from $A$ to $B$.

²Further Notation: Also written $\textup{Hom}_{\mathsf{Sets}}\webleft (A,B\webright )$.

Let $A$ and $B$ be sets.

Functoriality. The assignments $X,Y,\webleft (X,Y\webright )\mapsto \textup{Hom}_{\mathsf{Sets}}\webleft (X,Y\webright )$ define functors
\[ \begin{array}{ccc} \mathsf{Sets}\webleft (X,-\webright )\colon \mkern -15mu & \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets},\\ \mathsf{Sets}\webleft (-,Y\webright )\colon \mkern -15mu & \mathsf{Sets}^{\mathrlap {\mathsf{op}}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets},\\ \mathsf{Sets}\webleft (-_{1},-_{2}\webright )\colon \mkern -15mu & \mathsf{Sets}^{\mathsf{op}}\times \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}. \end{array} \]
Adjointness. We have adjunctions
witnessed by bijections
\begin{align*} \mathsf{Sets}\webleft (A\times B,C\webright ) & \cong \mathsf{Sets}\webleft (A,\mathsf{Sets}\webleft (B,C\webright )\webright ),\\ \mathsf{Sets}\webleft (A\times B,C\webright ) & \cong \mathsf{Sets}\webleft (B,\mathsf{Sets}\webleft (A,C\webright )\webright ), \end{align*}

natural in $A,B,C\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Maps From the Punctual Set. We have a bijection
\[ \mathsf{Sets}\webleft (\text{pt},A\webright )\cong A, \]

natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.
Maps to the Punctual Set. We have a bijection
\[ \mathsf{Sets}\webleft (A,\text{pt}\webright )\cong \text{pt}, \]

natural in $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$.

Item 1: Functoriality

This follows from Chapter 9: Categories, Item 2 and Item 5 of Proposition 9.1.4.1.2.

Item 2: Adjointness

This is a repetition of Item 2 of Proposition 2.1.3.1.3 and is proved there.

Item 3: Maps From the Punctual Set

The bijection

\[ \Phi _{A}\colon \mathsf{Sets}\webleft (\text{pt},A\webright )\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }A \]

is given by

\[ \Phi _{A}\webleft (f\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (\star \webright ) \]

for each $f\in \mathsf{Sets}\webleft (\text{pt},A\webright )$, admitting an inverse

\[ \Phi ^{-1}_{A}\colon A\overset {\scriptstyle \mathord {\sim }}{\dashrightarrow }\mathsf{Sets}\webleft (\text{pt},A\webright ) \]

given by

\[ \Phi ^{-1}_{A}\webleft (a\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[\star \mapsto a]\mspace {-3mu}] \]

for each $a\in A$. Indeed, we have

\begin{align*} \webleft [\Phi ^{-1}_{A}\circ \Phi _{A}\webright ]\webleft (f\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi ^{-1}_{A}\webleft (\Phi _{A}\webleft (f\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi ^{-1}_{A}\webleft (f\webleft (\star \webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[\mspace {-3mu}[\star \mapsto f\webleft (\star \webright )]\mspace {-3mu}]\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\text{id}_{\mathsf{Sets}\webleft (\text{pt},A\webright )}\webright ]\webleft (f\webright ) \end{align*}

for each $f\in \mathsf{Sets}\webleft (\text{pt},A\webright )$ and

\begin{align*} \webleft [\Phi _{A}\circ \Phi ^{-1}_{A}\webright ]\webleft (a\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{A}\webleft (\Phi ^{-1}_{A}\webleft (a\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{A}\webleft ([\mspace {-3mu}[\star \mapsto a]\mspace {-3mu}]\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathrm{ev}_{\star }\webleft ([\mspace {-3mu}[\star \mapsto a]\mspace {-3mu}]\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}a\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\text{id}_{A}\webright ]\webleft (a\webright ) \end{align*}

for each $a\in A$, and thus we have

\begin{align*} \Phi ^{-1}_{A}\circ \Phi _{A} & = \text{id}_{\mathsf{Sets}\webleft (\text{pt},A\webright )}\\ \Phi _{A}\circ \Phi ^{-1}_{A} & = \text{id}_{A}. \end{align*}

To prove naturality, we need to show that the diagram

commutes. Indeed, we have

\begin{align*} \webleft [f\circ \Phi _{A}\webright ]\webleft (\phi \webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (\Phi _{A}\webleft (\phi \webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (\phi \webleft (\star \webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [f\circ \phi \webright ]\webleft (\star \webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{B}\webleft (f\circ \phi \webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\Phi _{B}\webleft (f_{*}\webleft (\phi \webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\Phi _{B}\circ f_{*}\webright ]\webleft (\phi \webright ) \end{align*}

for each $\phi \in \mathsf{Sets}\webleft (\text{pt},A\webright )$. This finishes the proof.

Item 4: Maps to the Punctual Set

This follows from the universal property of $\text{pt}$ as the terminal set, Definition 2.1.1.1.1.

The Clowder Project

2.3.5 Sets of Maps