3 Monoidal Structures on the Category of Sets

This chapter contains some material on monoidal structures on $\mathsf{Sets}$.

• Section 3.1: The Monoidal Category of Sets and Products
  • Subsection 3.1.1: Products of Sets
  • Subsection 3.1.2: The Internal Hom of Sets
  • Subsection 3.1.3: The Monoidal Unit
    • Definition 3.1.3.1.1: The Monoidal Unit of $\times $
  • Subsection 3.1.4: The Associator
    • Definition 3.1.4.1.1: The Associator of $\times $
  • Subsection 3.1.5: The Left Unitor
    • Definition 3.1.5.1.1: The Left Unitor of $\times $
  • Subsection 3.1.6: The Right Unitor
    • Definition 3.1.6.1.1: The Right Unitor of $\times $
  • Subsection 3.1.7: The Symmetry
    • Definition 3.1.7.1.1: The Symmetry of $\times $
  • Subsection 3.1.8: The Diagonal
    • Definition 3.1.8.1.1: The Diagonal of $\times $
    • Proposition 3.1.8.1.2: Properties of the Diagonal Map
  • Subsection 3.1.9: The Monoidal Category of Sets and Products
    • Proposition 3.1.9.1.1: The Monoidal Structure on Sets Associated to the Product
  • Subsection 3.1.10: The Universal Property of $\webleft (\mathsf{Sets},\times ,\text{pt}\webright )$
    • Theorem 3.1.10.1.1: The Universal Property of $\webleft (\mathsf{Sets},\times ,\text{pt}\webright )$
    • Corollary 3.1.10.1.2: A Second Universal Property for $\webleft (\mathsf{Sets},\times ,\text{pt}\webright )$
• Section 3.2: The Monoidal Category of Sets and Coproducts
  • Subsection 3.2.1: Coproducts of Sets
  • Subsection 3.2.2: The Monoidal Unit
    • Definition 3.2.2.1.1: The Monoidal Unit of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
  • Subsection 3.2.3: The Associator
    • Definition 3.2.3.1.1: The Associator of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
  • Subsection 3.2.4: The Left Unitor
    • Definition 3.2.4.1.1: The Left Unitor of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
  • Subsection 3.2.5: The Right Unitor
    • Definition 3.2.5.1.1: The Right Unitor of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
  • Subsection 3.2.6: The Symmetry
    • Definition 3.2.6.1.1: The Symmetry of $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
  • Subsection 3.2.7: The Monoidal Category of Sets and Coproducts
    • Proposition 3.2.7.1.1: The Monoidal Structure on Sets Associated to $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
• Section 3.3: The Bimonoidal Category of Sets, Products, and Coproducts
  • Subsection 3.3.1: The Left Distributor
    • Definition 3.3.1.1.1: The Left Distributor of $\times $ over $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
  • Subsection 3.3.2: The Right Distributor
    • Definition 3.3.2.1.1: The Right Distributor of $\times $ over $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$
  • Subsection 3.3.3: The Left Annihilator
    • Definition 3.3.3.1.1: The Left Annihilator of $\times $
  • Subsection 3.3.4: The Right Annihilator
    • Definition 3.3.4.1.1: The Right Annihilator of $\times $
  • Subsection 3.3.5: The Bimonoidal Category of Sets, Products, and Coproducts
    • Proposition 3.3.5.1.1: The Bimonoidal Structure on Sets Associated to $\times $ and $\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}$