Item 9 (02TW) — The Clowder Project

8.7.2 Coinverse Images

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

The coinverse image function associated to $R$ is the function

\[ R_{-1}\colon \mathcal{P}(Y)\to \mathcal{P}(X) \]

defined by¹

\[ R_{-1}(V) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in X\ \middle |\ R(a)\subset V\right\} \]

for each $V\in \mathcal{P}(Y)$.

¹Further Terminology: The set $R_{-1}(V)$ is called the coinverse image of $V$ by $R$.

Remark 8.7.2.1.2 ▶ Unwinding Definition 8.7.2.1.1

Identifying $\mathcal{P}(Y)$ with $\mathrm{Rel}(\mathrm{pt},Y)$ via Chapter 4: Constructions With Sets, Item 3 of Proposition 4.4.1.1.4, we see that the inverse image function associated to $R$ is equivalently the function

\[ R_{-1}\colon \underbrace{\mathcal{P}(Y)}_{\cong \mathrm{Rel}(\mathrm{pt},Y)}\to \underbrace{\mathcal{P}(X)}_{\cong \mathrm{Rel}(\mathrm{pt},X)} \]

defined by

and being explicitly computed by

\begin{align*} R_{-1}(V) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Rift}}_{R}(V)\\ & \cong \int _{b\in Y}\operatorname {\mathrm{Hom}}_{\{ \mathsf{t},\mathsf{f}\} }(R^{b}_{-_{1}},V^{b}_{-_{2}}),\end{align*}

via Proposition 8.5.18.1.2.

Proof of Remark 8.7.2.1.2.

Indeed, we have

\begin{align*} \operatorname {\mathrm{Rift}}_{R}(V)& \cong \int _{b\in Y}\operatorname {\mathrm{Hom}}_{\{ \mathsf{t},\mathsf{f}\} }(R^{b}_{-_{1}},V^{b}_{-_{2}})\\ & =\left\{ a\in X\ \middle |\ \int _{b\in Y}\operatorname {\mathrm{Hom}}_{\{ \mathsf{t},\mathsf{f}\} }(R^{b}_{a},V^{b}_{\star })=\mathsf{true}\right\} \\ & = \left\{ a\in X\ \middle |\ \begin{aligned} & \text{For each $b\in Y$, at least one of the}\\ & \text{following conditions hold:}\\[2.5pt]& \mspace {25mu}\rlap {\text{1.}}\mspace {22.5mu}\text{We have $R^{b}_{a}=\mathsf{false}$.}\\ & \mspace {25mu}\rlap {\text{2.}}\mspace {22.5mu}\text{The following conditions hold:}\\[5.0pt]& \mspace {50mu}\rlap {\text{(a)}}\mspace {30mu}\text{We have $R^{b}_{a}=\mathsf{true}$.}\\ & \mspace {50mu}\rlap {\text{(b)}}\mspace {30mu}\text{We have $V^{b}_{\star }=\mathsf{true}$.}\\ & {}\end{aligned} \right\} \\ & = \left\{ a\in X\ \middle |\ \begin{aligned} & \text{For each $b\in Y$, at least one of the}\\ & \text{following conditions hold:}\\[2.5pt]& \mspace {25mu}\rlap {\text{1.}}\mspace {22.5mu}\text{We have $b\not\in R(a)$.}\\ & \mspace {25mu}\rlap {\text{2.}}\mspace {22.5mu}\text{The following conditions hold:}\\[7.5pt]& \mspace {50mu}\rlap {\text{(a)}}\mspace {30mu}\text{We have $b\in R(a)$.}\\ & \mspace {50mu}\rlap {\text{(b)}}\mspace {30mu}\text{We have $b\in V$.}\\ & {}\end{aligned} \right\} \\ & = \left\{ a\in X\ \middle |\ \text{for each $b\in R(a)$, we have $b\in V$}\right\} \\ & = \left\{ a\in X\ \middle |\ R(a)\subset V\right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1}(V).\end{align*}

This finishes the proof.

Definition 8.7.2.1.3 ▶ The Domain and Complement Parts of $R_{-1}$

Let $V$ be a subset of $Y$.¹

1.
The domain part of the coinverse image $R_{-1}(V)$ of $V$ is the set $R^{\mathrm{dom}}_{-1}(U)$ defined by

\begin{align*} R^{\mathrm{dom}}_{-1}(V) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1}(V)\cap \operatorname {Dom}(R)\\ & = \left\{ a\in X\ \middle |\ \begin{aligned} & \text{we have $R(a)\subset V$}\\ & \text{and $R(a)\neq \text{Ø}$.} \end{aligned}\right\} .\end{align*}
2.
The complement part of the coinverse image $R_{-1}(V)$ of $V$ is the set $R^{\mathrm{cp}}_{-1}(V)$ defined by

\begin{align*} R^{\mathrm{cp}}_{-}(V) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1}(V)\cap (X\setminus \operatorname {Dom}(R))\\ & = X\setminus \operatorname {Dom}(R)\\ & = \left\{ a\in X\ \middle |\ \begin{aligned} & \text{we have $R(a)\subset V$}\\ & \text{and $R(a)=\text{Ø}$.} \end{aligned}\right\} \\ & = \left\{ a\in X\ \middle |\ R(a)=\text{Ø}\right\} .\end{align*}

¹Note that we have
\[ R_{-1}(V)=R^{\mathrm{dom}}_{-1}(V)\cup R^{\mathrm{cp}}_{-1}(V), \]
as
\begin{align*} R_{-1}(V) & = R_{-1}(V)\cap X\\ & = R_{-1}(V)\cap (\operatorname {Dom}(R)\cup (X\setminus \operatorname {Dom}(R)))\\ & = (R_{-1}(V)\cap \operatorname {Dom}(R))\cup (R_{-1}(V)\cap (X\setminus \operatorname {Dom}(R)))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R^{\mathrm{dom}}_{-1}(V)\cup R^{\mathrm{cp}}_{-1}(V). \end{align*}
for each $V\in \mathcal{P}(Y)$.

Proposition 8.7.2.1.4 ▶ Properties of Coinverse Images I

Let $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y$ be a relation.

1.
Functoriality. The assignment $V\mapsto R_{-1}(V)$ defines a functor

\[ R_{-1}\colon (\mathcal{P}(Y),\subset )\to (\mathcal{P}(X),\subset ). \]

In particular, for each $U,V\in \mathcal{P}(Y)$, the following condition is satisfied:
- (★)
2.
Adjointness. We have an adjunction
witnessed by:
1. (a)
  Units and counits of the form
  
  \begin{align*} \operatorname {\mathrm{id}}_{\mathcal{P}(X)} \hookrightarrow R_{-1}\circ R_{!},\\ R_{!}\circ R_{-1} \hookrightarrow \operatorname {\mathrm{id}}_{\mathcal{P}(Y)}. \end{align*}
  
  In particular:
  - •
    For each $U\in \mathcal{P}(X)$, we have $U\subset R_{-1}(R_{!}(U))$.
  - •
    For each $V\in \mathcal{P}(Y)$, we have $R_{!}(R_{-1}(V))\subset V$.
2. (b)
  A bijections of sets
  
  \[ \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(R_{!}(U),V)\cong \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(U,R_{-1}(V)), \]
  
  natural in $U\in \mathcal{P}(X)$ and $V\in \mathcal{P}(Y)$. In particular:
  - (★)
3.
Interaction With Unions of Families of Subsets. The diagram
commutes, i.e. we have

\[ \bigcup _{V\in \mathcal{V}}R_{-1}(V)=\bigcup _{U\in R_{-1}(\mathcal{U})}U \]

for each $\mathcal{V}\in \mathcal{P}(Y)$, where $R_{-1}(\mathcal{V})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(R_{-1})_{!}(\mathcal{V})$.
4.
Interaction With Intersections of Families of Subsets. The diagram
commutes, i.e. we have

\[ \bigcap _{V\in \mathcal{V}}R_{-1}(V)=\bigcap _{U\in R_{-1}(\mathcal{U})}U \]

for each $\mathcal{V}\in \mathcal{P}(Y)$, where $R_{-1}(\mathcal{V})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(R_{-1})_{!}(\mathcal{V})$.
5.
Interaction With Binary Unions. We have a natural transformation
with components

\[ R_{-1}(U)\cup R_{-1}(V)\subset R_{-1}(U\cup V) \]

indexed by $U,V\in \mathcal{P}(Y)$.
6.
Interaction With Binary Intersections. The diagram
commutes, i.e. we have

\[ R_{-1}(U\cap V)=R_{-1}(U)\cap R_{-1}(V) \]

for each $U,V\in \mathcal{P}(Y)$.
7.
Interaction With Differences. We have a natural transformation
with components

\[ R_{-1}(U\setminus V)\subset R_{-1}(U)\setminus R_{-1}(V) \]

indexed by $U,V\in \mathcal{P}(Y)$.
8.
Interaction With Complements. The diagram
commutes, i.e. we have

\[ R_{-1}(U^{\textsf{c}})=R^{-1}(U)^{\textsf{c}} \]

for each $U\in \mathcal{P}(X)$.

9.
Interaction With Symmetric Differences. The diagram
does not commute in general, i.e. we may have

\[ R_{-1}(U)\mathbin {\triangle }R_{-1}(V)\neq R_{-1}(U\mathbin {\triangle }V) \]

in general, where $U,V\in \mathcal{P}(Y)$.

10.

Interaction With Internal Homs of Powersets. We have a natural transformation

with components

\[ [R^{-1}(U),R_{-1}(V)]_{X}\subset R_{-1}([U,V]_{Y}), \]

indexed by $U,V\in \mathcal{P}(Y)$.

11.

Lax Preservation of Colimits. We have an inclusion of sets

\[ \bigcup _{i\in I}R_{-1}(U_{i})\subset R_{-1}\left(\bigcup _{i\in I}U_{i}\right), \]

natural in $\left\{ U_{i}\right\} _{i\in I}\in \mathcal{P}(Y)^{\times I}$. In particular, we have inclusions

\[ \begin{gathered} R_{-1}(U)\cup R_{-1}(V) \subset R_{-1}(U\cup V),\\ \text{Ø}\subset R_{-1}(\text{Ø}), \end{gathered} \]

natural in $U,V\in \mathcal{P}(Y)$.

12.

Preservation of Limits. We have an equality of sets

\[ R_{-1}\left(\bigcap _{i\in I}U_{i}\right)=\bigcap _{i\in I}R_{-1}(U_{i}), \]

natural in $\left\{ U_{i}\right\} _{i\in I}\in \mathcal{P}(Y)^{\times I}$. In particular, we have equalities

\[ \begin{gathered} R_{-1}(U\cap V) = R_{-1}(U)\cap R_{-1}(V),\\ R_{-1}(Y) = Y, \end{gathered} \]

natural in $U,V\in \mathcal{P}(Y)$.

13.

Symmetric Lax Monoidality With Respect to Unions. The codirect image function of Item 1 has a symmetric lax monoidal structure

\[ (R_{-1},R^{\otimes }_{-1},R^{\otimes }_{-1|\mathbb {1}}) \colon (\mathcal{P}(X),\cup ,\text{Ø}) \to (\mathcal{P}(Y),\cup ,\text{Ø}), \]

being equipped with inclusions

\[ \begin{gathered} R^{\otimes }_{-1|U,V} \colon R_{-1}(U)\cup R_{-1}(V) \subset R_{-1}(U\cup V),\\ R^{\otimes }_{-1|\mathbb {1}} \colon \text{Ø}\subset R_{-1}(\text{Ø}), \end{gathered} \]

natural in $U,V\in \mathcal{P}(Y)$.

14.

Symmetric Strict Monoidality With Respect to Intersections. The direct image function of Item 1 has a symmetric strict monoidal structure

\[ (R_{-1},R^{\otimes }_{-1},R^{\otimes }_{-1|\mathbb {1}}) \colon (\mathcal{P}(X),\cap ,X) \to (\mathcal{P}(Y),\cap ,Y), \]

being equipped with equalities

\[ \begin{gathered} R^{\otimes }_{-1|U,V} \colon R_{-1}(U\cap V) \mathbin {\overset {=}{\rightarrow }}R_{-1}(U)\cap R_{-1}(V),\\ R^{\otimes }_{-1|\mathbb {1}} \colon R_{-1}(X) \mathbin {\overset {=}{\rightarrow }}Y, \end{gathered} \]

natural in $U,V\in \mathcal{P}(Y)$.

15.

Interaction With Coproducts. Let $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X'$ and $S\colon Y\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y'$ be relations. The diagram

commutes, i.e. we have

\[ (R\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}S)_{-1}(U'\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}V')=R_{-1}(U')\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}S_{-1}(V') \]

for each $U'\in \mathcal{P}(X')$ and each $V'\in \mathcal{P}(Y')$.

16.

Interaction With Products. Let $f\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X'$ and $g\colon Y\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y'$ be relations. The diagram

commutes, i.e. we have

\[ (R\boxtimes _{X'\times Y'}S)_{-1}(U'\boxtimes _{X'\times Y'}V')=R_{-1}(U')\boxtimes _{X\times Y}S_{-1}(V') \]

for each $U'\in \mathcal{P}(X')$ and each $V'\in \mathcal{P}(Y')$.

17.

Relation to Inverse Images. We have

\[ R_{-1}(V)=X\setminus R^{-1}(Y\setminus V) \]

for each $V\in \mathcal{P}(Y)$.

18.

Interaction With Functional Relations. If $R$ is functional, then we have

\begin{align*} R^{\mathrm{dom}}_{-1}(V) & = R^{-1}(V),\\ R^{\mathrm{cp}}_{-1}(V) & = X\setminus \operatorname {Dom}(R),\end{align*}

\begin{align*} R_{-1}(V) & = R^{\mathrm{dom}}_{-1}(V)\cup R^{\mathrm{cp}}_{-1}(V)\\ & = R^{-1}(V)\cup (X\setminus \operatorname {Dom}(R))\end{align*}

for each $V\in \mathcal{P}(Y)$.

19.

Interaction With Total Relations. If $R$ is total, then we have

\begin{align*} R^{\mathrm{dom}}_{-1}(V) & \subset R^{-1}(V),\\ R^{\mathrm{cp}}_{-1}(V) & = \text{Ø},\end{align*}

\[ R_{-1}(V)\subset R^{-1}(V) \]

for each $V\in \mathcal{P}(Y)$.

20.

Interaction With Functions I. If $R$ is a function, then we have

\begin{align*} R^{\mathrm{dom}}_{-1}(V) & = R^{-1}(V),\\ R^{\mathrm{cp}}_{-1}(V) & = \text{Ø},\end{align*}

so $R_{-1}=R^{-1}$.

21.

Interaction With Functions II. If $R_{-1}=R^{-1}$, then $R$ is a function.

Proof of Proposition 8.7.2.1.4.

Item 1: Functoriality

Omitted.

Item 2: Adjointness

This follows from

Item 3: Interaction With Unions of Families of Subsets

We have

\begin{align*} \bigcup _{U\in R_{-1}(\mathcal{V})}U & = \bigcup _{U\in \left\{ R_{-1}(V)\in \mathcal{P}(X)\ \middle |\ V\in \mathcal{V}\right\} }U\\ & = \bigcup _{V\in \mathcal{V}}R_{-1}(V).\end{align*}

This finishes the proof.

Item 4: Interaction With Intersections of Families of Subsets

We have

\begin{align*} \bigcap _{U\in R_{-1}(\mathcal{V})}U & = \bigcap _{U\in \left\{ R_{-1}(V)\in \mathcal{P}(X)\ \middle |\ V\in \mathcal{V}\right\} }U\\ & = \bigcap _{V\in \mathcal{V}}R_{-1}(V).\end{align*}

This finishes the proof.

Item 5: Interaction With Binary Unions

We have

\begin{align*} R_{-1}(U)\cup R_{-1}(V) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in X\ \middle |\ R(a)\subset U\right\} \cup \left\{ a\in X\ \middle |\ R(a)\subset V\right\} \\ & = \left\{ a\in X\ \middle |\ \text{$R(a)\subset U$ or $R(a)\subset V$}\right\} \\ & \subset \left\{ a\in X\ \middle |\ R(a)\subset U\cup V\right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1}(U\cup V). \end{align*}

Since $\mathcal{P}(X)$ is posetal, naturality is automatic (Chapter 11: Categories, Item 4 of Proposition 11.2.7.1.2). This finishes the proof.

Item 6: Interaction With Binary Intersections

We have

\begin{align*} R_{-1}(U)\cap R_{-1}(V) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in X\ \middle |\ R(a)\subset U\right\} \cap \left\{ a\in X\ \middle |\ R(a)\subset V\right\} \\ & = \left\{ a\in X\ \middle |\ \text{$R(a)\subset U$ and $R(a)\subset V$}\right\} \\ & = \left\{ a\in X\ \middle |\ R(a)\subset U\cap V\right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1}(U\cap V). \end{align*}

This finishes the proof.

Item 7: Interaction With Differences

We have

\begin{align*} R_{-1}(U\setminus V) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in X\ \middle |\ R(a)\subset U\setminus V\right\} \\ & \subset \left\{ a\in X\ \middle |\ \text{$R(a)\subset U$ and $R(a)\centernot {\subset }V$}\right\} \\ & = \left\{ a\in X\ \middle |\ R(a)\subset U\right\} \setminus \left\{ a\in X\ \middle |\ R(a)\subset V\right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1}(U)\setminus R_{-1}(V)\\ \end{align*}

Since $\mathcal{P}(X)$ is posetal, naturality is automatic (Chapter 11: Categories, Item 4 of Proposition 11.2.7.1.2). This finishes the proof.

Item 8: Interaction With Complements

Applying

to $X\setminus U$, we have

\begin{align*} R_{-1}(U^{\textsf{c}}) & = R_{-1}(X\setminus U)\\ & = Y\setminus R^{-1}(X\setminus (X\setminus U))\\ & = Y\setminus R^{-1}(U)\\ & = R^{-1}(U)^{\textsf{c}}. \end{align*}

This finishes the proof.

Item 9: Interaction With Symmetric Differences

Omitted.

Item 10: Interaction With Internal Homs of Powersets

We have

\begin{align*} [R^{-1}(U),R_{-1}(V)]_{X} & = R^{-1}(U)^{\textsf{c}}\cup R_{-1}(V)\\ & = R_{-1}(U^{\textsf{c}})\cup R_{-1}(V)\\ & \subset R_{-1}(U^{\textsf{c}}\cup V)\\ & = R_{-1}([U,V]_{Y}).\end{align*}

Since $\mathcal{P}(X)$ is posetal, naturality is automatic (Chapter 11: Categories, Item 4 of Proposition 11.2.7.1.2). This finishes the proof.

Item 11: Lax Preservation of Colimits

Omitted.

Item 12: Preservation of Limits

This follows from Item 2 and

Item 13: Symmetric Lax Monoidality With Respect to Unions

This follows from Item 11.

Item 14: Symmetric Strict Monoidality With Respect to Intersections

This follows from Item 12.

Item 15: Interaction With Coproducts

Omitted.

Item 16: Interaction With Products

Omitted.

Item 17: Relation to Inverse Images

We claim we have an equality

\[ R_{-1}(Y\setminus V)=X\setminus R^{-1}(V). \]

Indeed, we have

\begin{align*} R_{-1}(Y\setminus V) & = \left\{ a\in X\ |\ R(a)\subset Y\setminus V\right\} ,\\ X\setminus R^{-1}(V) & = \left\{ a\in X\ |\ R(a)\cap V=\text{Ø}\right\} .\end{align*}

Taking $V=Y\setminus V$ then implies the original statement.

Item 18: Interaction With Functional Relations

When $a\in \operatorname {Dom}(R)$, the set $R(a)$ will have exactly one element, so the condition $R(a)\cap V\neq \text{Ø}$ will then be equivalent to $R(a)\subset V$. Thus $R^{\mathrm{dom}}_{-1}(V)=R^{-1}(V)$.

Item 19: Interaction With Total Relations

Indeed, we have

\begin{align*} R^{\mathrm{cp}}_{-1}(V) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}X\setminus \operatorname {Dom}(R)\\ & = X\setminus X\\ & = \text{Ø}. \end{align*}

Since $R(a)\subset V$ is a strictly stronger condition than $R(a)\cap V\neq \text{Ø}$ when $R(a)$ is nonempty, we have $R^{\mathrm{dom}}_{-1}(V)\subset R^{-1}(V)$. Thus $R_{-1}(V)\subset R^{-1}(V)$.

Item 20: Interaction With Functions I

This follows from Item 18 and Item 19.

Item 21: Interaction With Functions II

This follows from Item 6 of Proposition 8.2.2.1.2.

Proposition 8.7.2.1.5 ▶ Properties of Coinverse Images II

Let $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y$ be a relation.

1.
Functionality I. The assignment $R\mapsto R_{-1}$ defines a function

\[ (-)_{-1}\colon \mathsf{Sets}(X,Y) \to \mathsf{Sets}(\mathcal{P}(X),\mathcal{P}(Y)). \]
2.
Functionality II. The assignment $R\mapsto R_{-1}$ defines a function

\[ (-)_{-1}\colon \mathsf{Sets}(X,Y) \to \mathsf{Pos}((\mathcal{P}(X),\subset ),(\mathcal{P}(Y),\subset )). \]
3.
Interaction With Identities. For each $X\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, we have

\[ (\Delta _{X})_{-1}=\operatorname {\mathrm{id}}_{\mathcal{P}(X)}. \]
4.
Interaction With Composition. For each pair of composable relations $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y$ and $S\colon Y\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}C$, we have
5.
Interaction With Converses. We have

\[ R^{\dagger }_{-1}=R_{*}. \]

Proof of Proposition 8.7.2.1.5.

Item 1: Functionality I

There is nothing to prove.

Item 2: Functionality II

This follows from Item 1 of Proposition 8.7.2.1.4.

Item 3: Interaction With Identities

Indeed, we have

\begin{align*} (\Delta _{X})_{-1}(U) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in X\ \middle |\ \Delta _{X}(a)\subset U\right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in X\ \middle |\ \left\{ a\right\} \subset U\right\} \\ & = U \end{align*}

for each $U\in \mathcal{P}(X)$. Thus $(\Delta _{X})_{-1}=\operatorname {\mathrm{id}}_{\mathcal{P}(X)}$.

Item 4: Interaction With Composition

Indeed, we have

\begin{align*} (S\mathbin {\diamond }R)_{-1}(U) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in X\ \middle |\ [S\mathbin {\diamond }R](a)\subset U\right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in X\ \middle |\ S(R(a))\subset U\right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ a\in X\ \middle |\ S_{!}(R(a))\subset U\right\} \\ & = \left\{ a\in X\ \middle |\ R(a)\subset S_{-1}(U)\right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1}(S_{-1}(U))\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}[R_{-1}\circ S_{-1}](U) \end{align*}

for each $U\in \mathcal{P}(C)$, where we used Item 2 of Proposition 8.7.2.1.4, which implies that the conditions

•
We have $S_{!}(R(a))\subset U$.
•
We have $R(a)\subset S_{-1}(U)$.

are equivalent. Thus $(S\mathbin {\diamond }R)_{-1}=R_{-1}\circ S_{-1}$.

Item 5: Interaction With Converses

Omitted.

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