9 Constructions With Relations
This chapter contains some material about constructions with relations. Notably, we discuss and explore:
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1.
The existence or non-existence of Kan extensions and Kan lifts in the $2$-category $\boldsymbol {\mathsf{Rel}}$ (
).
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2.
The various kinds of constructions involving relations, such as graphs, domains, images, unions, intersections, products, converse relations, composition of relations, and collages (Section 9.2).
This chapter is under revision. TODO:
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1.
Co/limits in $\boldsymbol {\mathsf{Rel}}$.
- Section 9.1: Co/Limits in the Category of Relations
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Section 9.2: More Constructions With Relations
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Subsection 9.2.1: The Domain and Range of a Relation
- Definition 9.2.1.1.1: The Domain and Range of a Relation
- Subsection 9.2.2: Binary Unions of Relations
- Subsection 9.2.3: Unions of Families of Relations
- Subsection 9.2.4: Binary Intersections of Relations
- Subsection 9.2.5: Intersections of Families of Relations
- Subsection 9.2.6: Binary Products of Relations
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Subsection 9.2.7: Products of Families of Relations
- Definition 9.2.7.1.1: The Product of a Family of Relations
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Subsection 9.2.8: The Collage of a Relation
- Definition 9.2.8.1.1: The Collage of a Relation
- Notation 9.2.8.1.2: Notation: $\mathsf{Pos}_{/\Delta ^{1}}(A,B)$
- Remark 9.2.8.1.3: Unwinding Notation 9.2.8.1.2
- Proposition 9.2.8.1.4: Properties of Collages of Relations
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Subsection 9.2.1: The Domain and Range of a Relation
