The Clowder Project

6.2 Categories of Relations

• Subsection 6.2.1: The Category of Relations Between Two Sets
  • Definition 6.2.1.1.1: The Category of Relations Between Two Sets
• Subsection 6.2.2: The Category of Relations
  • Definition 6.2.2.1.1: The Category of Relations
• Subsection 6.2.3: The Closed Symmetric Monoidal Category of Relations
  • Subsubsection 6.2.3.1: The Monoidal Product
    • Definition 6.2.3.1.1: The Monoidal Product of $\mathsf{Rel}$
  • Subsubsection 6.2.3.2: The Monoidal Unit
    • Definition 6.2.3.2.1: The Monoidal Unit of $\mathsf{Rel}$
  • Subsubsection 6.2.3.3: The Associator
    • Definition 6.2.3.3.1: The Associator of $\mathsf{Rel}$
  • Subsubsection 6.2.3.4: The Left Unitor
    • Definition 6.2.3.4.1: The Left Unitor of $\mathsf{Rel}$
  • Subsubsection 6.2.3.5: The Right Unitor
    • Definition 6.2.3.5.1: The Right Unitor of $\mathsf{Rel}$
  • Subsubsection 6.2.3.6: The Symmetry
    • Definition 6.2.3.6.1: The Symmetry of $\mathsf{Rel}$
  • Subsubsection 6.2.3.7: The Internal Hom
    • Definition 6.2.3.7.1: The Internal Hom of $\mathsf{Rel}$
    • Proposition 6.2.3.7.2: Properties of the Internal Hom of $\mathsf{Rel}$
  • Subsubsection 6.2.3.8: The Closed Symmetric Monoidal Category of Relations
    • Proposition 6.2.3.8.1: The Closed Symmetric Monoidal Category of Relations
• Subsection 6.2.4: The $2$-Category of Relations
  • Definition 6.2.4.1.1: The $2$-Category of Relations
• Subsection 6.2.5: The Double Category of Relations
  • Subsubsection 6.2.5.1: The Double Category of Relations
    • Definition 6.2.5.1.1: The Double Category of Relations
  • Subsubsection 6.2.5.2: Horizontal Identities
    • Definition 6.2.5.2.1: The Horizontal Identities of $\mathsf{Rel}^{\mathsf{dbl}}$
  • Subsubsection 6.2.5.3: Horizontal Composition
    • Definition 6.2.5.3.1: The Horizontal Composition of $\mathsf{Rel}^{\mathsf{dbl}}$
  • Subsubsection 6.2.5.4: Vertical Composition of 2-Morphisms
    • Definition 6.2.5.4.1: The Vertical Composition of 2-Morphisms in $\mathsf{Rel}^{\mathsf{dbl}}$
  • Subsubsection 6.2.5.5: The Associators
    • Definition 6.2.5.5.1: The Associators of $\mathsf{Rel}^{\mathsf{dbl}}$
  • Subsubsection 6.2.5.6: The Left Unitors
    • Definition 6.2.5.6.1: The Left Unitors of $\mathsf{Rel}^{\mathsf{dbl}}$
  • Subsubsection 6.2.5.7: The Right Unitors
    • Definition 6.2.5.7.1: The Right Unitors of $\mathsf{Rel}^{\mathsf{dbl}}$