7.3.8 The Left Skew Monoidal Structure on Pointed Sets Associated to $\lhd $

Proposition 7.3.8.1.1 ▶ The Left Skew Monoidal Structure on Pointed Sets Associated to $\lhd $

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

  • •

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

  • •

    The Left Skew Monoidal Product. The left tensor product functor

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

    of Definition 7.3.1.1.1.

  • •

    The Left Internal Skew Hom. The left internal Hom functor

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

    of Definition 7.3.2.1.1.

  • •

    The Left Skew Monoidal Unit. The functor

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

    of Definition 7.3.3.1.1.

  • •

    The Left Skew Associators. The natural transformation

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

    of Definition 7.3.4.1.1.

  • •

    The Left Skew Left Unitors. The natural transformation

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

    of Definition 7.3.5.1.1.

  • •

    The Left Skew Right Unitors. The natural transformation

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

    of Definition 7.3.6.1.1.

Proof of Proposition 7.3.8.1.1.

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 Left 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
and hence indeed commutes. Thus the left skew triangle identity is satisfied.

The Left 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 hence indeed commutes. Thus the right skew triangle identity is satisfied.

The Left 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.

Left Skew Monoidal Left-Closedness
This follows from Item 2 of Proposition 7.3.1.1.7.

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