7.4.8 The Right Skew Monoidal Structure on Pointed Sets Associated to $\rhd $

Proposition 7.4.8.1.1 ▶ The Right Skew Monoidal Structure on Pointed Sets Associated to $\rhd $

The category $\mathsf{Sets}_{*}$ admits a right-closed right skew monoidal category structure consisting of:

  • •

    The Underlying Category. The category $\mathsf{Sets}_{*}$ of pointed sets.

  • •

    The Right Skew Monoidal Product. The right tensor product functor

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

    of Definition 7.4.1.1.1.

  • •

    The Right Internal Skew Hom. The right internal Hom functor

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

    of Definition 7.4.2.1.1.

  • •

    The Right Skew Monoidal Unit. The functor

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

    of Definition 7.4.3.1.1.

  • •

    The Right Skew Associators. The natural transformation

    \[ \alpha ^{\mathsf{Sets}_{*},\rhd } \colon {\rhd }\circ {(\operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}}\times {\rhd })} \Longrightarrow {\rhd }\circ {({\rhd }\times \operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}})}\circ {\mathbf{\alpha }^{\mathsf{Cats},-1}_{\mathsf{Sets}_{*},\mathsf{Sets}_{*},\mathsf{Sets}_{*}}} \]

    of Definition 7.4.4.1.1.

  • •

    The Right Skew Left Unitors. The natural transformation

    \[ \lambda ^{\mathsf{Sets}_{*},\rhd }\colon \mathbf{\lambda }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}_{*}} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}{\rhd }\circ {(\mathbb {1}^{\mathsf{Sets}_{*}}\times \operatorname {\mathrm{id}}_{\mathsf{Sets}_{*}})} \]

    of Definition 7.4.5.1.1.

  • •

    The Right Skew Right Unitors. The natural transformation

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

    of Definition 7.4.6.1.1.

Proof of Proposition 7.4.8.1.1.

The Pentagon Identity
Let $(W,w_{0})$, $(X,x_{0})$, $(Y,y_{0})$ and $(Z,z_{0})$ 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 Right Skew Left Triangle Identity
Let $(X,x_{0})$ and $(Y,y_{0})$ be pointed sets. We have to show that the diagram
commutes. Indeed, this diagram acts on elements as
and hence indeed commutes. Thus the left skew triangle identity is satisfied.

The Right Skew Right Triangle Identity
Let $(X,x_{0})$ and $(Y,y_{0})$ be pointed sets. We have to show that the diagram
commutes. Indeed, this diagram acts on elements as
and
and hence indeed commutes. Thus the right skew triangle identity is satisfied.

The Right Skew Middle Triangle Identity
Let $(X,x_{0})$ and $(Y,y_{0})$ be pointed sets. We have to show that the diagram
commutes. Indeed, this diagram acts on elements as
and hence indeed commutes. Thus the right skew triangle identity is satisfied.

The Zig-Zag Identity
We have to show that the diagram
commutes. Indeed, this diagram acts on elements as
and
and hence indeed commutes. Thus the zig-zag identity is satisfied.

Right Skew Monoidal Right-Closedness
This follows from Item 2 of Proposition 7.4.1.1.7.

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