Subsection 5.5.2 (00GB): The Internal Hom of Pointed Sets

Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets.

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The internal Hom¹ of pointed sets from $\webleft (X,x_{0}\webright )$ to $\webleft (Y,y_{0}\webright )$ is the pointed set $\textbf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )$² consisting of:

The Underlying Set. The set $\mathsf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )$ of morphisms of pointed sets from $\webleft (X,x_{0}\webright )$ to $\webleft (Y,y_{0}\webright )$.
The Basepoint. The element

\[ \Delta _{y_{0}}\colon \webleft (X,x_{0}\webright )\to \webleft (Y,y_{0}\webright ) \]

of $\mathsf{Sets}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )$ given by

\[ \Delta _{y_{0}}\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}y_{0} \]

for each $x\in X$.

¹The pointed set $\textbf{Sets}_{*}\webleft (X,Y\webright )$ is the internal $\mathbf{Hom}$ of $\mathsf{Sets}_{*}$ with respect to the smash product of Chapter 5: Tensor Products of Pointed Sets, Definition 5.5.1.1.1; see Chapter 5: Tensor Products of Pointed Sets, Item 2 of Proposition 5.5.1.1.9.

²Further Notation: Also written $\mathbf{Hom}_{\textbf{Sets}_{*}}\webleft (X,Y\webright )$.

For a proof that $\textbf{Sets}_{*}$ is indeed the internal Hom of $\mathsf{Sets}_{*}$ with respect to the smash product of pointed sets, see Item 2 of Proposition 5.5.1.1.9.

Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets.

Functoriality. The assignments $X,Y,\webleft (X,Y\webright )\mapsto \textbf{Sets}_{*}\webleft (X,Y\webright )$ define functors
\[ \begin{array}{ccc} \textbf{Sets}_{*}\webleft (X,-\webright )\colon \mkern -15mu & \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ \textbf{Sets}_{*}\webleft (-,Y\webright )\colon \mkern -15mu & \mathsf{Sets}^{\mathrlap {\mathsf{op}}}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*},\\ \textbf{Sets}_{*}\webleft (-_{1},-_{2}\webright )\colon \mkern -15mu & \mathsf{Sets}^{\mathsf{op}}_{*}\times \mathsf{Sets}_{*} \mkern -17.5mu& {}\mathbin {\to }\mathsf{Sets}_{*}. \end{array} \]

In particular, given pointed maps

\begin{align*} f & \colon \webleft (X,x_{0}\webright ) \to \webleft (A,a_{0}\webright ),\\ g & \colon \webleft (Y,y_{0}\webright ) \to \webleft (B,b_{0}\webright ), \end{align*}

the induced map

\[ \textbf{Sets}_{*}\webleft (f,g\webright )\colon \textbf{Sets}_{*}\webleft (A,Y\webright )\to \textbf{Sets}_{*}\webleft (X,B\webright ) \]

is given by

\[ \webleft [\textbf{Sets}_{*}\webleft (f,g\webright )\webright ]\webleft (\phi \webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}g\circ \phi \circ f \]

for each $\phi \in \textbf{Sets}_{*}\webleft (A,Y\webright )$.
Adjointness. We have adjunctions
witnessed by bijections
\begin{align*} \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\wedge Y,Z\webright ) & \cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X,\textbf{Sets}_{*}\webleft (Y,Z\webright )\webright ),\\ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\wedge Y,Z\webright ) & \cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X,\textbf{Sets}_{*}\webleft (A,Z\webright )\webright ), \end{align*}

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$.
Enriched Adjointness. We have $\mathsf{Sets}_{*}$-enriched adjunctions
witnessed by isomorphisms of pointed sets
\begin{align*} \textbf{Sets}_{*}\webleft (X\wedge Y,Z\webright ) & \cong \textbf{Sets}_{*}\webleft (X,\textbf{Sets}_{*}\webleft (Y,Z\webright )\webright ),\\ \textbf{Sets}_{*}\webleft (X\wedge Y,Z\webright ) & \cong \textbf{Sets}_{*}\webleft (X,\textbf{Sets}_{*}\webleft (A,Z\webright )\webright ), \end{align*}

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\textbf{Sets}_{*}\webright )$.

Item 1: Functoriality

This follows from Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.5.1.2 and from the equalities

\begin{align*} g\circ \Delta _{y_{0}} & = \Delta _{z_{0}},\\ \Delta _{y_{0}}\circ f & = \Delta _{y_{0}} \end{align*}

for morphisms $f\colon \webleft (K,k_{0}\webright )\to \webleft (X,x_{0}\webright )$ and $g\colon \webleft (Y,y_{0}\webright )\to \webleft (Z,z_{0}\webright )$, which guarantee pre- and postcomposition by morphisms of pointed sets to also be morphisms of pointed sets.

Item 2: Adjointness

This is a repetition of Item 2 of Proposition 5.5.1.1.9, and is proved there.

Item 3: Enriched Adjointness

This is a repetition of Item 3 of Proposition 5.5.1.1.9, and is proved there.

The Clowder Project

5.5.2 The Internal Hom of Pointed Sets