Subsection 5.3.2 (00DQ): The Left 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 left internal Hom of pointed sets is the functor

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

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 5.2.2.1.4.

For a proof that $\webleft [-,-\webright ]^{\lhd }_{\mathsf{Sets}_{*}}$ is indeed the left internal Hom of $\mathsf{Sets}_{*}$ with respect to the left tensor product of pointed sets, see Item 2 of Proposition 5.3.1.1.7.

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

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

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

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

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

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

\begin{align*} \webleft [X,Y\webright ]^{\lhd }_{\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 $\bigwedge _{x\in X}\webleft (Y,y_{0}\webright )$.

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

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

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

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

is given by

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

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

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \text{Obj}\webleft (\mathsf{Sets}_{*}\webright )$
Adjointness II. The functor
\[ X\lhd -\colon \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]

does not admit a right adjoint.

Item 1: Functoriality

This follows from the definition of $\webleft [-,-\webright ]^{\lhd }_{\mathsf{Sets}_{*}}$ as a composition of functors (Definition 5.3.2.1.1).

Item 2: Adjointness I

This is a repetition of Item 2 of Proposition 5.3.1.1.7, and is proved there.

Item 3: Adjointness II

This is a repetition of Item 3 of Proposition 5.3.1.1.7, and is proved there.

The Clowder Project

5.3.2 The Left Internal Hom of Pointed Sets