The Clowder Project

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

• The Underlying Category. The category $\mathsf{Sets}$ of pointed sets.
• The Monoidal Product. The product functor
  \[ \times \colon \mathsf{Sets}\times \mathsf{Sets}\to \mathsf{Sets} \]

  of Chapter 2: Constructions With Sets, Item 1 of Proposition 2.1.3.1.3.

• The Internal Hom. The internal Hom functor
  \[ \mathsf{Sets}\colon \mathsf{Sets}^{\mathsf{op}}\times \mathsf{Sets}\to \mathsf{Sets} \]

  of Chapter 2: Constructions With Sets, Item 1 of Proposition 2.3.5.1.2.

• The Monoidal Unit. The functor
  \[ \mathbb {1}^{\mathsf{Sets}} \colon \mathsf{pt}\to \mathsf{Sets} \]

  of Definition 3.1.3.1.1.

• The Associators. The natural isomorphism
  \[ \alpha ^{\mathsf{Sets}} \colon {\times }\circ {\webleft ({\times }\times \text{id}_{\mathsf{Sets}}\webright )} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}{\times }\circ {\webleft (\text{id}_{\mathsf{Sets}}\times {\times }\webright )}\circ {\mathbf{\alpha }^{\mathsf{Cats}}_{\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}} \]

  of Definition 3.1.4.1.1.

• The Left Unitors. The natural isomorphism
  \[ \lambda ^{\mathsf{Sets}}\colon {\times }\circ {\webleft (\mathbb {1}^{\mathsf{Sets}}\times \text{id}_{\mathsf{Sets}}\webright )} \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathbf{\lambda }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}} \]

  of Definition 3.1.5.1.1.

• The Right Unitors. The natural isomorphism
  \[ \rho ^{\mathsf{Sets}}\colon {\times }\circ {\webleft ({\mathsf{id}}\times {\mathbb {1}^{\mathsf{Sets}}}\webright )}\mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}\mathbf{\rho }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}} \]

  of Definition 3.1.6.1.1.

• The Symmetry. The natural isomorphism
  \[ \sigma ^{\mathsf{Sets}} \colon {\times } \mathbin {\overset {\mathord {\sim }}{\Longrightarrow }}{\times }\circ {\mathbf{\sigma }^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets},\mathsf{Sets}}} \]

  of Definition 3.1.7.1.1.

• The Diagonals. The monoidal natural transformation
  \[ \Delta \colon \text{id}_{\mathsf{Sets}}\Longrightarrow \times \circ \Delta ^{\mathsf{Cats}_{\mathsf{2}}}_{\mathsf{Sets}} \]

  of Definition 3.1.8.1.1.