Subsection 7.4.6 (00TE): Functoriality of Powersets: Relations on Powersets

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Table of Contents
Part 2: Relations
Chapter 7: Constructions With Relations
Section 7.4: Functoriality of Powersets
Subsection 7.4.6: Functoriality of Powersets: Relations on Powersets (cite)

7.4.6 Functoriality of Powersets: Relations on Powersets

Let $A$ and $B$ be sets and let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.

Definition 7.4.6.1.1 ▶ The Relation on Powersets Associated to a Relation

The relation on powersets associated to $R$ is the relation

\[ \mathcal{P}\webleft (R\webright )\colon \mathcal{P}\webleft (A\webright )\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}\mathcal{P}\webleft (B\webright ) \]

defined by¹

\[ \mathcal{P}\webleft (R\webright )^{V}_{U}\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\mathbf{Rel}\webleft (\chi _{\text{pt}},V\mathbin {\diamond }R\mathbin {\diamond }U\webright ) \]

for each $U\in \mathcal{P}\webleft (A\webright )$ and each $V\in \mathcal{P}\webleft (B\webright )$.

¹Illustration:

Remark 7.4.6.1.2 ▶ Unwinding Definition 7.4.6.1.1

In detail, we have $U\sim _{\mathcal{P}\webleft (R\webright )}V$ iff the following equivalent conditions hold:

We have $\chi _{\text{pt}}\subset V\mathbin {\diamond }R\mathbin {\diamond }U$.
We have $\webleft (V\mathbin {\diamond }R\mathbin {\diamond }U\webright )^{\star }_{\star }=\mathsf{true}$, i.e. we have

\[ \int ^{a\in A}\int ^{b\in B}V^{\star }_{b}\times R^{b}_{a}\times U^{a}_{\star }=\mathsf{true}. \]
There exists some $a\in A$ and some $b\in B$ such that:
- We have $U^{a}_{\star }=\mathsf{true}$;
- We have $R^{b}_{a}=\mathsf{true}$;
- We have $V^{\star }_{b}=\mathsf{true}$.
There exists some $a\in A$ and some $b\in B$ such that:
- We have $a\in U$;
- We have $a\sim _{R}b$;
- We have $b\in V$.