Item 3 (00F6) — The Clowder Project

7.4.2 The Right Internal Hom of Pointed Sets

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

Definition 7.4.2.1.1 ▶ The Right Internal Hom of Pointed Sets

The right internal Hom¹ of pointed sets is the functor

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

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 7.2.2.1.4.

¹For a proof that $[-,-]^{\rhd }_{\mathsf{Sets}_{*}}$ is indeed the right internal Hom of $\mathsf{Sets}_{*}$ with respect to the right tensor product of pointed sets, see Item 2 of Proposition 7.4.1.1.7.

Remark 7.4.2.1.2 ▶ Unwinding Definition 7.4.2.1.1, I: Comparison With $\smash {[-,-]^{\lhd }_{\mathsf{Sets}_{*}}}$

We have

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

Remark 7.4.2.1.3 ▶ Unwinding Definition 7.4.2.1.1, II: Universal Property

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

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

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

1.
Pointed maps $f\colon X\rhd Y\to Z$.
2.
Pointed maps $f\colon Y\to [X,Z]^{\rhd }_{\mathsf{Sets}_{*}}$.

Remark 7.4.2.1.4 ▶ Unwinding Definition 7.4.2.1.1, III: Explicit Description

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

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

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

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

Proposition 7.4.2.1.5 ▶ Properties of Right Internal Homs of Pointed Sets

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

1.
Functoriality. The assignments $X,Y,(X,Y)\mapsto [X,Y]^{\rhd }_{\mathsf{Sets}_{*}}$ define functors

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

In particular, given pointed maps

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

the induced map

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

is given by

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

for each $[(y_{a})_{a\in A}]\in [A,Y]^{\rhd }_{\mathsf{Sets}_{*}}$.
2.
Adjointness I. We have an adjunction
witnessed by a bijection of sets

\[ \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(X\rhd Y,Z)\cong \operatorname {\mathrm{Hom}}_{\mathsf{Sets}_{*}}(Y,[X,Z]^{\rhd }_{\mathsf{Sets}_{*}}) \]

natural in $(X,x_{0}),(Y,y_{0}),(Z,z_{0})\in \operatorname {\mathrm{Obj}}(\mathsf{Sets}_{*})$, where $[X,Y]^{\rhd }_{\mathsf{Sets}_{*}}$ is the pointed set of Definition 7.4.2.1.1.