Remark 8.5.6.1.2 (00TG) — The Clowder Project

Table of Contents
Part 3: Relations
Chapter 8: Relations
Section 8.5: The Adjoint Pairs $R_{!}\dashv R_{-1}$ and $R^{-1}\dashv R_{*}$
Subsection 8.5.6: Functoriality of Powersets: Relations on Powersets
Remark 8.5.6.1.2: Unwinding Definition 8.5.6.1.1 (cite)

8.5.6 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 8.5.6.1.1 ▶ 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

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

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

¹Illustration:

Remark 8.5.6.1.2 ▶ Unwinding Definition 8.5.6.1.1

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

•
We have $\chi _{\mathrm{pt}}\subset V\mathbin {\diamond }R\mathbin {\diamond }U$.
•
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 $b\in V$.