Item 4 (00S9) — The Clowder Project

8.5.2 Strong Inverse Images

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

Definition 8.5.2.1.1 ▶ Strong Inverse Images

The strong inverse image function associated to $R$ is the function

\[ R_{-1}\colon \mathcal{P}\webleft (B\webright )\to \mathcal{P}\webleft (A\webright ) \]

defined by¹

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

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

¹Further Terminology: The set $R_{-1}\webleft (V\webright )$ is called the strong inverse image of $V$ by $R$.

Remark 8.5.2.1.2 ▶ Unwinding Definition 8.5.2.1.1

Identifying subsets of $B$ with relations from $\mathrm{pt}$ to $B$ via Chapter 4: Constructions With Sets, of , we see that the inverse image function associated to $R$ is equivalently the function

\[ R_{-1}\colon \underbrace{\mathcal{P}\webleft (B\webright )}_{\cong \mathrm{Rel}\webleft (\mathrm{pt},B\webright )}\to \underbrace{\mathcal{P}\webleft (A\webright )}_{\cong \mathrm{Rel}\webleft (\mathrm{pt},A\webright )} \]

defined by

and being explicitly computed by

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

where we have used .

Proof of Remark 8.5.2.1.2.

We have

\begin{align*} \operatorname {\mathrm{Rift}}_{R}\webleft (V\webright )& \cong \int _{b\in B}\operatorname {\mathrm{Hom}}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{b}_{-_{1}},V^{b}_{-_{2}}\webright )\\ & =\left\{ a\in A\ \middle |\ \int _{b\in B}\operatorname {\mathrm{Hom}}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{b}_{a},V^{b}_{\star }\webright )=\mathsf{true}\right\} \\ & = \left\{ a\in A\ \middle |\ \begin{aligned} & \text{for each $b\in B$, at least one of the}\\[-2.5pt]& \text{following conditions hold:}\\[7.5pt]& \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:}\\[7.5pt]& \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 $V^{b}_{\star }=\mathsf{true}$}\\[10pt]\end{aligned} \right\} \\ & = \left\{ a\in A\ \middle |\ \begin{aligned} & \text{for each $b\in B$, at least one of the}\\[-2.5pt]& \text{following conditions hold:}\\[7.5pt]& \mspace {25mu}\rlap {\text{1.}}\mspace {22.5mu}\text{We have $b\not\in R\webleft (a\webright )$}\\ & \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 $b\in R\webleft (a\webright )$}\\ & \mspace {50mu}\rlap {\text{(b)}}\mspace {30mu}\text{We have $b\in V$}\\[10pt]\end{aligned} \right\} \\ & = \left\{ a\in A\ \middle |\ \text{for each $b\in R\webleft (a\webright )$, we have $b\in V$}\right\} \\ & = \left\{ a\in A\ \middle |\ R\webleft (a\webright )\subset V\right\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1}\webleft (V\webright ).\end{align*}

This finishes the proof.

Proposition 8.5.2.1.3 ▶ Properties of Strong Inverse Images

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

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

\[ R_{-1}\colon \webleft (\mathcal{P}\webleft (B\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (A\webright ),\subset \webright ) \]

where
- •
  Action on Objects. For each $V\in \mathcal{P}\webleft (B\webright )$, we have
  
  \[ \webleft [R_{-1}\webright ]\webleft (V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1}\webleft (V\webright ). \]
- •
  Action on Morphisms. For each $U,V\in \mathcal{P}\webleft (B\webright )$:
  - •
    If $U\subset V$, then $R_{-1}\webleft (U\webright )\subset R_{-1}\webleft (V\webright )$.
2.
Adjointness. We have an adjunction
witnessed by a bijections of sets

\[ \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (A\webright )}\webleft (R_{!}\webleft (U\webright ),V\webright )\cong \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (A\webright )}\webleft (U,R_{-1}\webleft (V\webright )\webright ), \]

natural in $U\in \mathcal{P}\webleft (A\webright )$ and $V\in \mathcal{P}\webleft (B\webright )$, i.e. such that:
- (★)
3.
Lax Preservation of Colimits. We have an inclusion of sets

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

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

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

natural in $U,V\in \mathcal{P}\webleft (B\webright )$.
4.
Preservation of Limits. We have an equality of sets

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

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

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

natural in $U,V\in \mathcal{P}\webleft (B\webright )$.
5.
Symmetric Lax Monoidality With Respect to Unions. The codirect image function of Item 1 has a symmetric lax monoidal structure

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

being equipped with inclusions

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

natural in $U,V\in \mathcal{P}\webleft (B\webright )$.
6.
Symmetric Strict Monoidality With Respect to Intersections. The direct image function of Item 1 has a symmetric strict monoidal structure

\[ \webleft (R_{-1},R^{\otimes }_{-1},R^{\otimes }_{-1|\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (A\webright ),\cap ,A\webright ) \to \webleft (\mathcal{P}\webleft (B\webright ),\cap ,B\webright ), \]

being equipped with equalities

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

natural in $U,V\in \mathcal{P}\webleft (B\webright )$.
7.
Interaction With Weak Inverse Images I. We have

\[ R_{-1}\webleft (V\webright )=A\setminus R^{-1}\webleft (B\setminus V\webright ) \]

for each $V\in \mathcal{P}\webleft (B\webright )$.
8.
Interaction With Weak Inverse Images II. Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation from $A$ to $B$.
1. (a)
  If $R$ is a total relation, then we have an inclusion of sets
  
  \[ R_{-1}\webleft (V\webright ) \subset R^{-1}\webleft (V\webright ) \]
  
  natural in $V\in \mathcal{P}\webleft (B\webright )$.
2. (b)
  If $R$ is total and functional, then the above inclusion is in fact an equality.
3. (c)
  Conversely, if we have $R_{-1}=R^{-1}$, then $R$ is total and functional.

Proof of Proposition 8.5.2.1.3.

Item 1: Functoriality

Clear.

Item 2: Adjointness

This follows from

Item 3: Lax Preservation of Colimits

Omitted.

Item 4: Preservation of Limits

This follows from Item 2 and

Item 5: Symmetric Lax Monoidality With Respect to Unions

This follows from Item 3.

Item 6: Symmetric Strict Monoidality With Respect to Intersections

This follows from Item 4.

Item 7: Interaction With Weak Inverse Images I

We claim we have an equality

\[ R_{-1}\webleft (B\setminus V\webright )=A\setminus R^{-1}\webleft (V\webright ). \]

Indeed, we have

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

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

Item 8: Interaction With Weak Inverse Images II

Item 8a is clear, while Item 8b and Item 8c follow from Item 6 of Proposition 8.2.2.1.2.

Proposition 8.5.2.1.4 ▶ Properties of the Strong Inverse Image Function Operation

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

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

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

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

\[ \webleft (\operatorname {\mathrm{id}}_{A}\webright )_{-1}=\operatorname {\mathrm{id}}_{\mathcal{P}\webleft (A\webright )}. \]

4.
Interaction With Composition. For each pair of composable relations $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ and $S\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}C$, we have

Proof of Proposition 8.5.2.1.4.

Item 1: Functionality I

Clear.

Item 2: Functionality II

Clear.

Item 3: Interaction With Identities

Indeed, we have

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

for each $U\in \mathcal{P}\webleft (A\webright )$. Thus $\webleft (\chi _{A}\webright )_{-1}=\operatorname {\mathrm{id}}_{\mathcal{P}\webleft (A\webright )}$.

Item 4: Interaction With Composition

Indeed, we have

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

for each $U\in \mathcal{P}\webleft (C\webright )$, where we used Item 2 of Proposition 8.5.2.1.3, which implies that the conditions

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

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

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