The Clowder Project

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

• The Underlying Category. The category $\mathsf{Sets}$ of pointed sets.
• The Monoidal Product. The coproduct functor
  $$\coprod :\mathsf{Sets}\times \mathsf{Sets}\to \mathsf{Sets}$$

  of Chapter 2: Constructions With Sets, Item 1 of Proposition 2.2.3.1.3.

• The Monoidal Unit. The functor
  $${\mathbb{0}}^{\mathsf{Sets}}:\mathsf{pt}\to \mathsf{Sets}$$

  of Definition 3.2.2.1.1.

• The Associators. The natural isomorphism
  $${\alpha }^{\mathsf{Sets},\coprod }:\coprod \circ (\coprod \times {\text{id}}_{\mathsf{Sets}})\stackrel{\sim }{⟹}\coprod \circ ({\text{id}}_{\mathsf{Sets}}\times \coprod )\circ {\alpha }_{\mathsf{Sets},\mathsf{Sets},\mathsf{Sets}}^{\mathsf{Cats}}$$

  of Definition 3.2.3.1.1.

• The Left Unitors. The natural isomorphism
  $${\lambda }^{\mathsf{Sets},\coprod }:\coprod \circ ({\mathbb{0}}^{\mathsf{Sets}}\times {\text{id}}_{\mathsf{Sets}})\stackrel{\sim }{⟹}{\lambda }_{\mathsf{Sets}}^{{\mathsf{Cats}}_{\mathsf{2}}}$$

  of Definition 3.2.4.1.1.

• The Right Unitors. The natural isomorphism
  $${\rho }^{\mathsf{Sets},\coprod }:\coprod \circ (\mathsf{id}\times {\mathbb{0}}^{\mathsf{Sets}})\stackrel{\sim }{⟹}{\rho }_{\mathsf{Sets}}^{{\mathsf{Cats}}_{\mathsf{2}}}$$

  of Definition 3.2.5.1.1.

• The Symmetry. The natural isomorphism
  $${\sigma }^{\mathsf{Sets},\coprod }:\times \stackrel{\sim }{⟹}\times \circ {\sigma }_{\mathsf{Sets},\mathsf{Sets}}^{{\mathsf{Cats}}_{\mathsf{2}}}$$

  of Definition 3.2.6.1.1.