Subsection 4.4.2 (00EY): The Right Internal Hom of Pointed Sets

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4.4.2 The Right Internal Hom of Pointed Sets

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

Definition 4.4.2.1.1 ▶ The Right Internal Hom of Pointed Sets

The right internal Hom of pointed sets is the functor

\[ \webleft [-,-\webright ]^{\rhd }_{\mathsf{Sets}_{*}}\colon \mathsf{Sets}^{\mathsf{op}}_{*}\times \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]

defined as the composition

\[ \mathsf{Sets}^{\mathsf{op}}_{*}\times \mathsf{Sets}_{*}\overset {{\text{忘}}\times \mathsf{id}}{\to }\mathsf{Sets}^{\mathsf{op}}\times \mathsf{Sets}_{*}\overset {\pitchfork }{\to }\mathsf{Sets}_{*}, \]

where:

${\text{忘}}\colon \mathsf{Sets}_{*}\to \mathsf{Sets}$ is the forgetful functor from pointed sets to sets.
$\pitchfork \colon \mathsf{Sets}^{\mathsf{op}}\times \mathsf{Sets}_{*}\to \mathsf{Sets}_{*}$ is the cotensor functor of Item 1 of Proposition 4.2.2.1.4.

Proof of Definition 4.4.2.1.1.

For a proof that $\webleft [-,-\webright ]^{\rhd }_{\mathsf{Sets}_{*}}$ is indeed the right internal Hom of $\mathsf{Sets}_{*}$ with respect to the right tensor product of pointed sets, see Item 2 of Proposition 4.4.1.1.7.

Remark 4.4.2.1.2 ▶ Unwinding Definition 4.4.2.1.1, I: Comparison With $\smash {\webleft [-,-\webright ]^{\lhd }_{\mathsf{Sets}_{*}}}$

We have

\[ \webleft [-,-\webright ]^{\lhd }_{\mathsf{Sets}_{*}}=\webleft [-,-\webright ]^{\rhd }_{\mathsf{Sets}_{*}}. \]

Remark 4.4.2.1.3 ▶ Unwinding Definition 4.4.2.1.1, II: Universal Property

The right internal Hom of pointed sets satisfies the following universal property:

\[ \mathsf{Sets}_{*}\webleft (X\rhd Y,Z\webright )\cong \mathsf{Sets}_{*}\webleft (Y,\webleft [X,Z\webright ]^{\rhd }_{\mathsf{Sets}_{*}}\webright ) \]

That is to say, the following data are in bijection:

Pointed maps $f\colon X\rhd Y\to Z$.
Pointed maps $f\colon Y\to \webleft [X,Z\webright ]^{\rhd }_{\mathsf{Sets}_{*}}$.

Remark 4.4.2.1.4 ▶ Unwinding Definition 4.4.2.1.1, III: Explicit Description

In detail, the right internal Hom of $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ is the pointed set $\smash {\webleft (\webleft [X,Y\webright ]^{\rhd }_{\mathsf{Sets}_{*}},\webleft [\webleft (y_{0}\webright )_{x\in X}\webright ]\webright )}$ consisting of

The Underlying Set. The set $\webleft [X,Y\webright ]^{\rhd }_{\mathsf{Sets}_{*}}$ defined by

\begin{align*} \webleft [X,Y\webright ]^{\rhd }_{\mathsf{Sets}_{*}} & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\lvert X\right\rvert \pitchfork Y\\ & \cong \bigwedge _{x\in X}\webleft (Y,y_{0}\webright ), \end{align*}

where $\left\lvert X\right\rvert $ denotes the underlying set of $\webleft (X,x_{0}\webright )$;
The Underlying Basepoint. The point $\webleft [\webleft (y_{0}\webright )_{x\in X}\webright ]$ of $\bigwedge _{x\in X}\webleft (Y,y_{0}\webright )$.

Proposition 4.4.2.1.5 ▶ Properties of Right Internal Homs of Pointed Sets

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

Functoriality. The assignments $X,Y,\webleft (X,Y\webright )\mapsto \webleft [X,Y\webright ]^{\rhd }_{\mathsf{Sets}_{*}}$ define functors
\begin{gather*} \begin{aligned} \webleft [X,-\webright ]^{\rhd }_{\mathsf{Sets}_{*}} & \colon \mathsf{Sets}_{*} \to \mathsf{Sets}_{*},\\ \webleft [-,Y\webright ]^{\rhd }_{\mathsf{Sets}_{*}} & \colon \mathsf{Sets}^{\mathsf{op}}_{*} \to \mathsf{Sets}_{*},\\ \end{aligned}\\ \webleft [-_{1},-_{2}\webright ]^{\rhd }_{\mathsf{Sets}_{*}} \colon \mathsf{Sets}^{\mathsf{op}}_{*}\times \mathsf{Sets}_{*} \to \mathsf{Sets}_{*}. \end{gather*}

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

\[ \webleft [f,g\webright ]^{\rhd }_{\mathsf{Sets}_{*}}\colon \webleft [A,Y\webright ]^{\rhd }_{\mathsf{Sets}_{*}}\to \webleft [X,B\webright ]^{\rhd }_{\mathsf{Sets}_{*}} \]

is given by

\[ \webleft [f,g\webright ]^{\rhd }_{\mathsf{Sets}_{*}}\webleft (\webleft [\webleft (y_{a}\webright )_{a\in A}\webright ]\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [\webleft (g\webleft (y_{f\webleft (x\webright )}\webright )\webright )_{x\in X}\webright ] \]

for each $\webleft [\webleft (y_{a}\webright )_{a\in A}\webright ]\in \webleft [A,Y\webright ]^{\rhd }_{\mathsf{Sets}_{*}}$.
Adjointness I. We have an adjunction
witnessed by a bijection of sets
\[ \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (X\rhd Y,Z\webright )\cong \textup{Hom}_{\mathsf{Sets}_{*}}\webleft (Y,\webleft [X,Z\webright ]^{\rhd }_{\mathsf{Sets}_{*}}\webright ) \]

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$, where $\webleft [X,Y\webright ]^{\rhd }_{\mathsf{Sets}_{*}}$ is the pointed set of Definition 4.4.2.1.1.
Adjointness II. The functor
\[ -\rhd Y\colon \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]

does not admit a right adjoint.