6 Relations

This chapter contains some material about relations. Notably, we discuss and explore:

1. 1. The definition of relations (Section 6.1.1).
2. 2. How relations may be viewed as decategorification of profunctors (Section 6.1.2).
3. 3. The various kind of categories that relations form, namely:
   1. (a) A category (Section 6.2.2).
   2. (b) A monoidal category (Section 6.2.3).
   3. (c) A $2$-category (Section 6.2.4).
   4. (d) A double category (Section 6.2.5).
4. 4. The various categorical properties of the $2$-category of relations, including:
   1. (a) The self-duality of $\mathsf{Rel}$ and $\mathbf{\text{Rel}}$ (Proposition 6.3.1.1.1).
   2. (b) Identifications of equivalences and isomorphisms in $\mathbf{\text{Rel}}$ with bijections (Proposition 6.3.2.1.1).
   3. (c) Identifications of adjunctions in $\mathbf{\text{Rel}}$ with functions (Proposition 6.3.3.1.1).
   4. (d) Identifications of monads in $\mathbf{\text{Rel}}$ with preorders (Proposition 6.3.4.1.1).
   5. (e) Identifications of comonads in $\mathbf{\text{Rel}}$ with subsets (Proposition 6.3.5.1.1).
   6. (f) A description of the monoids and comonoids in $\mathbf{\text{Rel}}$ with respect to the Cartesian product (Remark 6.3.6.1.1).
   7. (g) Characterisations of monomorphisms in $\mathsf{Rel}$ (Proposition 6.3.7.1.1).
   8. (h) Characterisations of $2$-categorical notions of monomorphisms in $\mathbf{\text{Rel}}$ (Proposition 6.3.8.1.1).
   9. (i) Characterisations of epimorphisms in $\mathsf{Rel}$ (Proposition 6.3.9.1.1).
   10. (j) Characterisations of $2$-categorical notions of epimorphisms in $\mathbf{\text{Rel}}$ (Proposition 6.3.10.1.1).
   11. (k) The partial co/completeness of $\mathsf{Rel}$ (Proposition 6.3.11.1.1).
   12. (l) The existence or non-existence of Kan extensions and Kan lifts in $\mathsf{Rel}$ (Remark 6.3.12.1.1).
   13. (m) The closedness of $\mathbf{\text{Rel}}$ (Proposition 6.3.13.1.1).
   14. (n) The identification of $\mathbf{\text{Rel}}$ with the category of free algebras of the powerset monad on $\mathsf{Sets}$ (Proposition 6.3.14.1.1).
5. 5. A description of two notions of “skew composition” on $\mathbf{Rel}(A,B)$, giving rise to left and right skew monoidal structures analogous to the left skew monoidal structure on $\mathsf{Fun}(\mathcal{C},\mathcal{D})$ appearing in the definition of a relative monad (Section 6.4 and Section 6.5).

• Section 6.1: Relations
  • Subsection 6.1.1: Foundations
    • Definition 6.1.1.1.1: Relations
    • Remark 6.1.1.1.2: Unwinding Enumi 1, I
    • Remark 6.1.1.1.3: Unwinding Enumi 2, II
    • Notation 6.1.1.1.4: Further Notation for Relations
    • Proposition 6.1.1.1.5: Properties of Relations
  • Subsection 6.1.2: Relations as Decategorifications of Profunctors
    • Remark 6.1.2.1.1: Relations as Decategorifications of Profunctors I
    • Remark 6.1.2.1.2: Relations as Decategorifications of Profunctors II
  • Subsection 6.1.3: Examples of Relations
    • Example 6.1.3.1.1: The Trivial Relation
    • Example 6.1.3.1.2: The Cotrivial Relation
    • Example 6.1.3.1.3: The Characteristic Relation of a Set
    • Example 6.1.3.1.4: Square Roots
    • Example 6.1.3.1.5: Complex Logarithms
    • Example 6.1.3.1.6: More Examples of Relations
  • Subsection 6.1.4: Functional Relations
    • Definition 6.1.4.1.1: Functional Relations
    • Proposition 6.1.4.1.2: Properties of Functional Relations
  • Subsection 6.1.5: Total Relations
    • Definition 6.1.5.1.1: Total Relations
    • Proposition 6.1.5.1.2: Properties of Total Relations
• Section 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 ${\mathsf{Rel}}^{\mathsf{dbl}}$
• Section 6.3: Properties of the $2$-Category of Relations
  • Subsection 6.3.1: Self-Duality
    • Proposition 6.3.1.1.1: Self-Duality for the (2-)Category of Relations
  • Subsection 6.3.2: Isomorphisms and Equivalences in $\mathbf{\text{Rel}}$
    • Proposition 6.3.2.1.1: Isomorphisms and Equivalences in $\mathbf{\text{Rel}}$
  • Subsection 6.3.3: Adjunctions in $\mathbf{\text{Rel}}$
    • Proposition 6.3.3.1.1: Adjunctions in $\mathbf{\text{Rel}}$
  • Subsection 6.3.4: Monads in $\mathbf{\text{Rel}}$
    • Proposition 6.3.4.1.1: Monads in $\mathbf{\text{Rel}}$
  • Subsection 6.3.5: Comonads in $\mathbf{\text{Rel}}$
    • Proposition 6.3.5.1.1: Comonads in $\mathbf{\text{Rel}}$
  • Subsection 6.3.6: Co/Monoids in $\mathbf{\text{Rel}}$
    • Remark 6.3.6.1.1: Co/Monoids in $\mathbf{\text{Rel}}$
  • Subsection 6.3.7: Monomorphisms in $\mathsf{Rel}$
    • Proposition 6.3.7.1.1: Characterisations of Monomorphisms in $\mathsf{Rel}$
  • Subsection 6.3.8: 2-Categorical Monomorphisms in $\mathbf{\text{Rel}}$
    • Proposition 6.3.8.1.1: 2-Categorical Monomorphisms in $\mathbf{\text{Rel}}$
    • Question 6.3.8.1.2: Better Characterisations of Representably Full Morphisms in $\mathbf{\text{Rel}}$
  • Subsection 6.3.9: Epimorphisms in $\mathsf{Rel}$
    • Proposition 6.3.9.1.1: Characterisations of Epimorphisms in $\mathsf{Rel}$
  • Subsection 6.3.10: 2-Categorical Epimorphisms in $\mathbf{\text{Rel}}$
    • Proposition 6.3.10.1.1: 2-Categorical Epimorphisms in $\mathbf{\text{Rel}}$
    • Question 6.3.10.1.2: Better Characterisations of Corepresentably Full Morphisms in $\mathbf{\text{Rel}}$
  • Subsection 6.3.11: Co/Limits in $\mathsf{Rel}$
    • Proposition 6.3.11.1.1: Co/Limits in $\mathsf{Rel}$
  • Subsection 6.3.12: Kan Extensions and Kan Lifts in $\mathbf{\text{Rel}}$
    • Remark 6.3.12.1.1: Kan Extensions and Kan Lifts in $\mathbf{\text{Rel}}$
  • Subsection 6.3.13: Closedness of $\mathbf{\text{Rel}}$
    • Proposition 6.3.13.1.1: Closedness of $\mathbf{\text{Rel}}$
  • Subsection 6.3.14: $\mathbf{\text{Rel}}$ as a Category of Free Algebras
    • Proposition 6.3.14.1.1: $\mathbf{\text{Rel}}$ as a Category of Free Algebras
• Section 6.4: The Left Skew Monoidal Structure on $\mathbf{Rel}(A,B)$
  • Subsection 6.4.1: The Left Skew Monoidal Product
    • Definition 6.4.1.1.1: The Left $J$-Skew Monoidal Product of $\mathbf{Rel}(A,B)$
  • Subsection 6.4.2: The Left Skew Monoidal Unit
    • Definition 6.4.2.1.1: The Left $J$-Skew Monoidal Unit of $\mathbf{Rel}(A,B)$
  • Subsection 6.4.3: The Left Skew Associators
    • Definition 6.4.3.1.1: The Left $J$-Skew Associator of $\mathbf{Rel}(A,B)$
  • Subsection 6.4.4: The Left Skew Left Unitors
    • Definition 6.4.4.1.1: The Left $J$-Skew Left Unitor of $\mathbf{Rel}(A,B)$
  • Subsection 6.4.5: The Left Skew Right Unitors
    • Definition 6.4.5.1.1: The Left $J$-Skew Right Unitor of $\mathbf{Rel}(A,B)$
  • Subsection 6.4.6: The Left Skew Monoidal Structure on $\mathbf{Rel}(A,B)$
    • Proposition 6.4.6.1.1: The Left $J$-Skew Monoidal Structure on $\mathbf{Rel}(A,B)$
• Section 6.5: The Right Skew Monoidal Structure on $\mathbf{Rel}(A,B)$
  • Subsection 6.5.1: The Right Skew Monoidal Product
    • Definition 6.5.1.1.1: The Right $J$-Skew Monoidal Product of $\mathbf{Rel}(A,B)$
  • Subsection 6.5.2: The Right Skew Monoidal Unit
    • Definition 6.5.2.1.1: The Right $J$-Skew Monoidal Unit of $\mathbf{Rel}(A,B)$
  • Subsection 6.5.3: The Right Skew Associators
    • Definition 6.5.3.1.1: The Right $J$-Skew Associator of $\mathbf{Rel}(A,B)$
  • Subsection 6.5.4: The Right Skew Left Unitors
    • Definition 6.5.4.1.1: The Right $J$-Skew Left Unitor of $\mathbf{Rel}(A,B)$
  • Subsection 6.5.5: The Right Skew Right Unitors
    • Definition 6.5.5.1.1: The Right $J$-Skew Right Unitor of $\mathbf{Rel}(A,B)$
  • Subsection 6.5.6: The Right Skew Monoidal Structure on $\mathbf{Rel}(A,B)$
    • Proposition 6.5.6.1.1: The Right $J$-Skew Monoidal Structure on $\mathbf{Rel}(A,B)$