Item 1 (00GE) — The Clowder Project

7.5.2 The Internal Hom of Pointed Sets

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

Definition 7.5.2.1.1 ▶ The Internal Hom of Pointed Sets

The internal Hom¹ of pointed sets from $\webleft (X,x_{0}\webright )$ to $\webleft (Y,y_{0}\webright )$ is the pointed set $\boldsymbol {\mathsf{Sets}}_{*}\webleft (\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright )\webright )$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 $x\in X$.

¹For a proof that $\boldsymbol {\mathsf{Sets}}_{*}$ is indeed the internal Hom of $\mathsf{Sets}_{*}$ with respect to the smash product of pointed sets, see Item 2 of Proposition 7.5.1.1.10.
²Further Notation: Also written $\mathbf{Hom}_{\boldsymbol {\mathsf{Sets}}_{*}}\webleft (X,Y\webright )$.

Proposition 7.5.2.1.2 ▶ Properties of the Internal Hom of Pointed Sets

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

1.
Functoriality. The assignments $X,Y,\webleft (X,Y\webright )\mapsto \boldsymbol {\mathsf{Sets}}_{*}\webleft (X,Y\webright )$ define functors

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

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

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

is given by

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

for each $\phi \in \boldsymbol {\mathsf{Sets}}_{*}\webleft (A,Y\webright )$.

Adjointness. We have adjunctions

witnessed by bijections

\begin{align*} \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}\webleft (X\wedge Y,Z\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}\webleft (X,\boldsymbol {\mathsf{Sets}}_{*}\webleft (Y,Z\webright )\webright ),\\ \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}\webleft (X\wedge Y,Z\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}\webleft (X,\boldsymbol {\mathsf{Sets}}_{*}\webleft (A,Z\webright )\webright ), \end{align*}

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

Enriched Adjointness. We have $\mathsf{Sets}_{*}$-enriched adjunctions

witnessed by isomorphisms of pointed sets

\begin{align*} \boldsymbol {\mathsf{Sets}}_{*}\webleft (X\wedge Y,Z\webright ) & \cong \boldsymbol {\mathsf{Sets}}_{*}\webleft (X,\boldsymbol {\mathsf{Sets}}_{*}\webleft (Y,Z\webright )\webright ),\\ \boldsymbol {\mathsf{Sets}}_{*}\webleft (X\wedge Y,Z\webright ) & \cong \boldsymbol {\mathsf{Sets}}_{*}\webleft (X,\boldsymbol {\mathsf{Sets}}_{*}\webleft (A,Z\webright )\webright ), \end{align*}

natural in $\webleft (X,x_{0}\webright ),\webleft (Y,y_{0}\webright ),\webleft (Z,z_{0}\webright )\in \operatorname {\mathrm{Obj}}\webleft (\boldsymbol {\mathsf{Sets}}_{*}\webright )$.

Proof of Proposition 7.5.2.1.2.

Item 1: Functoriality

This follows from Chapter 4: Constructions With Sets, Item 1 of Proposition 4.3.5.1.2 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 7.5.1.1.10, and is proved there.

Item 3: Enriched Adjointness

This is a repetition of Item 3 of Proposition 7.5.1.1.10, and is proved there.

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