Definition 4.3.5.1.1 (003B): Sets of Maps

¹Further Terminology: Also called the Hom set from $A$ to $B$.
²Further Notation: Also written $\operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(A,B)$.

Let $A$ and $B$ be sets.

1.
Functoriality. The assignments $X,Y,(X,Y)\mapsto \operatorname {\mathrm{Hom}}_{\mathsf{Sets}}(X,Y)$ define functors

\[ \begin{array}{ccc} \mathsf{Sets}(X,-)\colon \mkern -15mu & \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets},\\ \mathsf{Sets}(-,Y)\colon \mkern -15mu & \mathsf{Sets}^{\mathrlap {\mathsf{op}}} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets},\\ \mathsf{Sets}(-_{1},-_{2})\colon \mkern -15mu & \mathsf{Sets}^{\mathsf{op}}\times \mathsf{Sets} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}. \end{array} \]
2.
Adjointness. We have adjunctions
witnessed by bijections

\begin{align*} \mathsf{Sets}(A\times B,C) & \cong \mathsf{Sets}(A,\mathsf{Sets}(B,C)),\\ \mathsf{Sets}(A\times B,C) & \cong \mathsf{Sets}(B,\mathsf{Sets}(A,C)), \end{align*}

natural in $A,B,C\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$.
3.
Maps From the Punctual Set. We have a bijection

\[ \mathsf{Sets}(\mathrm{pt},A)\cong A, \]

natural in $A\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$.
4.
Maps to the Punctual Set. We have a bijection

\[ \mathsf{Sets}(A,\mathrm{pt})\cong \mathrm{pt}, \]

natural in $A\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$.

Item 1: Functoriality

This follows from Chapter 11: Categories, Item 2 and Item 5 of Proposition 11.1.4.1.2.

Item 2: Adjointness

This is a repetition of Item 2 of Proposition 4.1.3.1.3 and is proved there.

Item 3: Maps From the Punctual Set

The bijection

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

is given by

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

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

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

given by

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

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

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

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

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

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

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

To prove naturality, we need to show that the diagram

commutes. Indeed, we have

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

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

Item 4: Maps to the Punctual Set

This follows from the universal property of $\mathrm{pt}$ as the terminal set, Definition 4.1.1.1.1.

The Clowder Project

4.3.5 Sets of Maps