4.5.13 Comonoids With Respect to the Smash Product of Pointed Sets

The symmetric monoidal functor

\[ \webleft (\webleft (-\webright )^{+},\webleft (-\webright )^{+,\times },\webleft (-\webright )^{+,\times }_{\mathbb {1}}\webright ) \colon \webleft (\mathsf{Sets},\times ,\text{pt}\webright ) \to \webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright ), \]

of Chapter 3: Pointed Sets, Item 4 of Proposition 3.4.1.1.2 lifts to an equivalence of categories

\begin{align*} \mathsf{CoMon}\webleft (\mathsf{Sets}_{*},\wedge ,S^{0}\webright ) & \mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{CoMon}\webleft (\mathsf{Sets},\times ,\text{pt}\webright )\\ & \cong \mathsf{Sets}. \end{align*}


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