Subsection 7.4.3 (00SA): Weak Inverse Images

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7.4.3 Weak Inverse Images

Let $A$ and $B$ be sets and let $R : A \to | B$ be a relation.

Definition 7.4.3.1.1 ▶ Weak Inverse Images

The weak inverse image function associated to $R$ ¹ is the function

R^{- 1} : P (B) \to P (A)

defined by²

R^{- 1} (V) \overset{def}{=} {a \in A | R (a) \cap V \neq Ø}

for each $V \in P (B)$ .

¹Further Terminology: Also called simply the inverse image function associated to $R$ .

²Further Terminology: The set

R^{- 1} (V)

is called the weak inverse image of $V$ by $R$ or simply the inverse image of $V$ by $R$ .

Remark 7.4.3.1.2 ▶ Unwinding Definition 7.4.3.1.1

Identifying subsets of $B$ with relations from $B$ to $pt$ via Chapter 2: Constructions With Sets, of , we see that the weak inverse image function associated to $R$ is equivalently the function

R^{- 1} : \underset{≅ Rel (B, pt)}{\underset{⏟}{P (B)}} \to \underset{≅ Rel (A, pt)}{\underset{⏟}{P (A)}}

defined by

R^{- 1} (V) \overset{def}{=} V ⋄ R

for each $V \in P (A)$ , where $R ⋄ V$ is the composition

A \overset{R}{\to |} B \overset{V}{\to |} pt .

Explicitly, we have

\begin{aligned} R^{- 1} (V) & \overset{def}{=} V ⋄ R \\ \overset{def}{=} \int^{x \in B} V_{x}^{-_{1}} \times R_{-_{2}}^{x} . \end{aligned}

Proof of Remark 7.4.3.1.2.

We have

\begin{aligned} V ⋄ R & \overset{def}{=} \int^{x \in B} V_{x}^{-_{1}} \times R_{-_{2}}^{x} \\ = {a \in A | \int^{x \in B} V_{x}^{⋆} \times R_{a}^{x} = true} \\ = {a \in A | \begin{aligned} there exists x \in B such that the \\ following conditions hold: \\ 1. We have V_{x}^{⋆} = true \\ 2. We have R_{a}^{x} = true \end{aligned}} \\ = {a \in A | \begin{aligned} there exists x \in B such that the \\ following conditions hold: \\ 1. We have x \in V \\ 2. We have x \in R (a) \end{aligned}} \\ = {a \in A | there exists x \in V such that x \in R (a)} \\ = {a \in A | R (a) \cap V \neq Ø} \\ \overset{def}{=} R^{- 1} (V) \end{aligned}

This finishes the proof.

Proposition 7.4.3.1.3 ▶ Properties of Weak Inverse Image Functions

Let $R : A \to | B$ be a relation.

1. Functoriality. The assignment $V \mapsto R^{- 1} (V)$ defines a functor
$R^{- 1} : (P (B), \subset) \to (P (A), \subset)$

where
- Action on Objects. For each $V \in P (B)$ , we have
  
  $[R^{- 1}] (V) \overset{def}{=} R^{- 1} (V);$
- Action on Morphisms. For each $U, V \in P (B)$ :
  - If $U \subset V$ , then $R^{- 1} (U) \subset R^{- 1} (V)$ .
2. Adjointness. We have an adjunction
witnessed by a bijections of sets
${Hom}_{P (A)} (R^{- 1} (U), V) ≅ {Hom}_{P (A)} (U, R_{!} (V)),$

natural in $U \in P (A)$ and $V \in P (B)$ , i.e. such that:
- The following conditions are equivalent:
  - We have $R^{- 1} (U) \subset V$ ;
  - We have $U \subset R_{!} (V)$ .
3. Preservation of Colimits. We have an equality of sets
$R^{- 1} (⋃_{i \in I} U_{i}) = ⋃_{i \in I} R^{- 1} (U_{i}),$

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

$\begin{matrix} R^{- 1} (U) \cup R^{- 1} (V) = R^{- 1} (U \cup V), \\ R^{- 1} (Ø) = Ø, \end{matrix}$

natural in $U, V \in P (B)$ .
4. Oplax Preservation of Limits. We have an inclusion of sets
$R^{- 1} (⋂_{i \in I} U_{i}) \subset ⋂_{i \in I} R^{- 1} (U_{i}),$

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

$\begin{matrix} R^{- 1} (U \cap V) \subset R^{- 1} (U) \cap R^{- 1} (V), \\ R^{- 1} (A) \subset B, \end{matrix}$

natural in $U, V \in P (B)$ .
5. Symmetric Strict Monoidality With Respect to Unions. The direct image function of Item 1 has a symmetric strict monoidal structure
$(R^{- 1}, R^{- 1, \otimes}, R_{1}^{- 1, \otimes}) : (P (A), \cup, Ø) \to (P (B), \cup, Ø),$

being equipped with equalities

$\begin{matrix} R_{U, V}^{- 1, \otimes} : R^{- 1} (U) \cup R^{- 1} (V) \overset{=}{\to} R^{- 1} (U \cup V), \\ R_{1}^{- 1, \otimes} : Ø \overset{=}{\to} Ø, \end{matrix}$

natural in $U, V \in P (B)$ .
6. Symmetric Oplax Monoidality With Respect to Intersections. The direct image function of Item 1 has a symmetric oplax monoidal structure
$(R^{- 1}, R^{- 1, \otimes}, R_{1}^{- 1, \otimes}) : (P (A), \cap, A) \to (P (B), \cap, B),$

being equipped with inclusions

$\begin{matrix} R_{U, V}^{- 1, \otimes} : R^{- 1} (U \cap V) \subset R^{- 1} (U) \cap R^{- 1} (V), \\ R_{1}^{- 1, \otimes} : R^{- 1} (A) \subset B, \end{matrix}$

natural in $U, V \in P (B)$ .
7. Interaction With Strong Inverse Images I. We have
$R^{- 1} (V) = A ∖ R_{- 1} (B ∖ V)$

for each $V \in P (B)$ .
8. Interaction With Strong Inverse Images II. Let $R : A \to | B$ be a relation from $A$ to $B$ .
1. (a) If $R$ is a total relation, then we have an inclusion of sets
  $R_{- 1} (V) \subset R^{- 1} (V)$
  
  natural in $V \in P (B)$ .
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 7.4.3.1.3.

Item 1: Functoriality

Clear.

Item 2: Adjointness

This follows from

Item 3: Preservation of Colimits

This follows from Item 2 and

Item 4: Oplax Preservation of Limits

Omitted.

Item 5: Symmetric Strict Monoidality With Respect to Unions

This follows from Item 3.

Item 6: Symmetric Oplax Monoidality With Respect to Intersections

This follows from Item 4.

Item 7: Interaction With Strong Inverse Images I

This follows from Item 7 of Proposition 7.4.2.1.3.

Item 8: Interaction With Strong Inverse Images II

This was proved in Item 8 of Proposition 7.4.2.1.3.

Proposition 7.4.3.1.4 ▶ Properties of the Weak Inverse Image Function Operation

Let $R : A \to | B$ be a relation.

1. Functionality I. The assignment $R \mapsto R^{- 1}$ defines a function
$(-)^{- 1} : Rel (A, B) \to Sets (P (A), P (B)) .$
2. Functionality II. The assignment $R \mapsto R^{- 1}$ defines a function
$(-)^{- 1} : Rel (A, B) \to Pos ((P (A), \subset), (P (B), \subset)) .$
3. Interaction With Identities. For each $A \in Obj (Sets)$ , we have¹
$(χ_{A})^{- 1} = {id}_{P (A)};$
4. Interaction With Composition. For each pair of composable relations $R : A \to | B$ and $S : B \to | C$ , we have²