9 Constructions With Relations

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

1. 1.

   The existence or non-existence of Kan extensions and Kan lifts in the $2$-category $\boldsymbol {\mathsf{Rel}}$ (Section 9.2).

2. 2.

   The various kinds of constructions involving relations, such as graphs, domains, ranges, unions, intersections, products, converse relations, composition of relations, and collages (Section 9.3).

• Section 9.1: Co/Limits in the Category of Relations
• Section 9.2: Kan Extensions and Kan Lifts in the $2$-Category of Relations
  • Subsection 9.2.1: Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
    • Proposition 9.2.1.1.1: Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
    • Question 9.2.1.1.2: Existence of Specific Left Kan Extensions of Relations
    • Question 9.2.1.1.3: Explicit Description of Left Kan Extensions Along Functions
  • Subsection 9.2.2: Left Kan Lifts in $\boldsymbol {\mathsf{Rel}}$
    • Proposition 9.2.2.1.1: Left Kan Lifts in $\boldsymbol {\mathsf{Rel}}$
    • Question 9.2.2.1.2: Existence of Specific Left Kan Lifts of Relations
    • Question 9.2.2.1.3: Explicit Description of Left Kan Lifts Along Functions
  • Subsection 9.2.3: Right Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
    • Proposition 9.2.3.1.1: Existence of Right Kan Extensions in $\boldsymbol {\mathsf{Rel}}$
  • Subsection 9.2.4: Right Kan Lifts in $\boldsymbol {\mathsf{Rel}}$
    • Proposition 9.2.4.1.1: Existence of Right Kan Lifts in $\boldsymbol {\mathsf{Rel}}$
• Section 9.3: More Constructions With Relations
  • Subsection 9.3.1: The Domain and Range of a Relation
    • Definition 9.3.1.1.1: The Domain and Range of a Relation
  • Subsection 9.3.2: Binary Unions of Relations
    • Definition 9.3.2.1.1: Binary Unions of Relations
    • Proposition 9.3.2.1.2: Properties of Binary Unions of Relations
  • Subsection 9.3.3: Unions of Families of Relations
    • Definition 9.3.3.1.1: The Union of a Family of Relations
    • Proposition 9.3.3.1.2: Properties of Unions of Families of Relations
  • Subsection 9.3.4: Binary Intersections of Relations
    • Definition 9.3.4.1.1: Binary Intersections of Relations
    • Proposition 9.3.4.1.2: Properties of Binary Intersections of Relations
  • Subsection 9.3.5: Intersections of Families of Relations
    • Definition 9.3.5.1.1: The Intersection of a Family of Relations
    • Proposition 9.3.5.1.2: Properties of Intersections of Families of Relations
  • Subsection 9.3.6: Binary Products of Relations
    • Definition 9.3.6.1.1: Binary Products of Relations
    • Proposition 9.3.6.1.2: Properties of Binary Products of Relations
  • Subsection 9.3.7: Products of Families of Relations
    • Definition 9.3.7.1.1: The Product of a Family of Relations
  • Subsection 9.3.8: The Collage of a Relation
    • Definition 9.3.8.1.1: The Collage of a Relation
    • Notation 9.3.8.1.2: Notation: $\mathsf{Pos}_{/\Delta ^{1}}\webleft (A,B\webright )$
    • Remark 9.3.8.1.3: Unwinding Notation 9.3.8.1.2
    • Proposition 9.3.8.1.4: Properties of Collages of Relations