This chapter contains some material about constructions with relations. Notably, we discuss and explore:
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The existence or non-existence of Kan extensions and Kan lifts in the $2$-category $\textbf{Rel}$ (Section 7.2).
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The various kinds of constructions involving relations, such as graphs, domains, ranges, unions, intersections, products, inverse relations, composition of relations, and collages (Section 7.3).
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The adjoint pairs
\begin{align*} R_{*} \dashv R_{-1} & \colon \mathcal{P}\webleft (A\webright ) \rightleftarrows \mathcal{P}\webleft (B\webright ),\\ R^{-1} \dashv R_{!} & \colon \mathcal{P}\webleft (B\webright ) \rightleftarrows \mathcal{P}\webleft (A\webright ) \end{align*}
of functors (morphisms of posets) between $\mathcal{P}\webleft (A\webright )$ and $\mathcal{P}\webleft (B\webright )$ induced by a relation $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$, as well as the properties of $R_{*}$, $R_{-1}$, $R^{-1}$, and $R_{!}$ (Section 7.4).
Of particular note are the following points:
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These two pairs of adjoint functors are the counterpart for relations of the adjoint triple $f_{*}\dashv f^{-1}\dashv f_{!}$ induced by a function $f\colon A\to B$ studied in Chapter 2: Constructions With Sets, Section 2.4;
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We have $R_{-1}=R^{-1}$ iff $R$ is total and functional (Item 8 of Proposition 7.4.2.1.3).
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As a consequence of the previous item, when $R$ comes from a function $f$, the pair of adjunctions
\[ R_{*}\dashv R_{-1}=R^{-1}\dashv R_{!} \]
reduces to the triple adjunction
\[ f_{*}\dashv f^{-1}\dashv f_{!} \]
from Chapter 2: Constructions With Sets, Section 2.4.
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The pairs $R_{*}\dashv R_{-1}$ and $R^{-1}\dashv R_{!}$ turn out to be rather important later on, as they appear in the definition and study of continuous, open, and closed relations between topological spaces (
, ).