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 5.5.1.1.10.
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 5.5.1.1.10.


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