Proposition 5.5.9.1.1 (00H1): The Monoidal Structure on Pointed Sets Associated to $\wedge $

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The category $\mathsf{Sets}_{*}$ admits a closed monoidal category with diagonals structure consisting of

The Underlying Category. The category $\mathsf{Sets}_{*}$ of pointed sets;
The Monoidal Product. The smash product functor

\[ \wedge \colon \mathsf{Sets}_{*}\times \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]

of Item 1 of Proposition 5.5.1.1.9;
The Internal Hom. The internal Hom functor

\[ \textbf{Sets}_{*}\colon \mathsf{Sets}^{\mathsf{op}}_{*}\times \mathsf{Sets}_{*}\to \mathsf{Sets}_{*} \]

of Item 1 of Proposition 5.5.2.1.2;
The Monoidal Unit. The functor

\[ \mathbb {1}^{\mathsf{Sets}_{*}} \colon \mathsf{pt}\to \mathsf{Sets}_{*} \]

of Definition 5.5.3.1.1;
The Associators. The natural isomorphism

\[ \alpha ^{\mathsf{Sets}_{*}} \colon {\wedge }\circ {\webleft ({\wedge }\times \text{id}_{\mathsf{Sets}_{*}}\webright )} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}{\wedge }\circ {\webleft (\text{id}_{\mathsf{Sets}_{*}}\times {\wedge }\webright )}\circ {\mathbf{\alpha }^{\mathsf{Cats}}_{\mathsf{Sets}_{*},\mathsf{Sets}_{*},\mathsf{Sets}_{*}}} \]

of Definition 5.5.4.1.1;
The Left Unitors. The natural isomorphism

\[ \lambda ^{\mathsf{Sets}_{*}}\colon {\wedge }\circ {\webleft (\mathbb {1}^{\mathsf{Sets}_{*}}\times \text{id}_{\mathsf{Sets}_{*}}\webright )} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathbf{\lambda }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*}} \]

of Definition 5.5.5.1.1;
The Right Unitors. The natural isomorphism

\[ \rho ^{\mathsf{Sets}_{*}}\colon {\wedge }\circ {\webleft ({\mathsf{id}}\times {\mathbb {1}^{\mathsf{Sets}_{*}}}\webright )}\mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathbf{\rho }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*}} \]

of Definition 5.5.6.1.1;
The Symmetry. The natural isomorphism

\[ \sigma ^{\mathsf{Sets}_{*}} \colon {\wedge } \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}{\wedge }\circ {\mathbf{\sigma }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*},\mathsf{Sets}_{*}}} \]

of Definition 5.5.7.1.1;
The Diagonals. The monoidal natural transformation

\[ \Delta ^{\wedge }\colon \text{id}_{\mathsf{Sets}_{*}}\Longrightarrow \wedge \circ \Delta ^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*}} \]

of Definition 5.5.8.1.1.

The Pentagon Identity

Let $\webleft (W,w_{0}\webright )$, $\webleft (X,x_{0}\webright )$, $\webleft (Y,y_{0}\webright )$ and $\webleft (Z,z_{0}\webright )$ be pointed sets. We have to show that the diagram

commutes. Indeed, this diagram acts on elements as

and thus we see that the pentagon identity is satisfied.

The Triangle Identity

Let $\webleft (X,x_{0}\webright )$ and $\webleft (Y,y_{0}\webright )$ be pointed sets. We have to show that the diagram

commutes. Indeed, this diagram acts on elements as

and

and thus we see that the triangle identity is satisfied.

The Left Hexagon Identity

Let $\webleft (X,x_{0}\webright )$, $\webleft (Y,y_{0}\webright )$, and $\webleft (Z,z_{0}\webright )$ be pointed sets. We have to show that the diagram

commutes. Indeed, this diagram acts on elements as

and thus we see that the left hexagon identity is satisfied.

The Right Hexagon Identity

Let $\webleft (X,x_{0}\webright )$, $\webleft (Y,y_{0}\webright )$, and $\webleft (Z,z_{0}\webright )$ be pointed sets. We have to show that the diagram

commutes. Indeed, this diagram acts on elements as

and thus we see that the right hexagon identity is satisfied.

Monoidal Closedness

This follows from Item 2 of Proposition 5.5.1.1.9.

Existence of Monoidal Diagonals

This follows from Item 1 and Item 2 of Proposition 5.5.8.1.2.