Proposition 4.6.1.1.5 (007K): Properties of Direct Images I

4.6.1 Direct Images

Let $f\colon X\to Y$ be a function.

The direct image function associated to $f$ is the function¹

\[ f_{!}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright ) \]

defined by²

\begin{align*} f_{!}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f\webleft (U\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ y\in Y\ \middle |\ \begin{aligned} & \text{there exists some $x\in U$}\\ & \text{such that $y=f\webleft (x\webright )$} \end{aligned} \right\} \\ & = \left\{ f\webleft (x\webright )\in Y\ \middle |\ x\in U\right\} \end{align*}

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

¹Further Notation: Also written simply $f\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright )$.
²Further Terminology: The set $f\webleft (U\webright )$ is called the direct image of $U$ by $f$.

Notation 4.6.1.1.2 ▶ Further Notation for Direct Images

Sometimes one finds the notation

\[ \exists _{f}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright ) \]

for $f_{!}$. This notation comes from the fact that the following statements are equivalent, where $y\in Y$ and $U\in \mathcal{P}\webleft (X\webright )$:

•
We have $y\in \exists _{f}\webleft (U\webright )$.
•
There exists some $x\in U$ such that $f\webleft (x\webright )=y$.

We will not make use of this notation elsewhere in Clowder.

ゴ

Warning 4.6.1.1.3 ▶ Notation for Direct Images Is Confusing

Notation for direct images between powersets is tricky:

1.
Direct images for powersets and presheaves are both adjoint to their corresponding inverse image functors. However, the direct image functor for powersets is a left adjoint, while the direct image functor for presheaves is a right adjoint:
1. (a)
  Powersets. Given a function $f\colon X\to Y$, we have an inverse image functor
  
  \[ f^{-1}\colon \mathcal{P}\webleft (Y\webright )\to \mathcal{P}\webleft (X\webright ). \]
  
  The left adjoint of this functor is the usual direct image, defined above in Definition 4.6.1.1.1.
2. (b)
  Presheaves. Given a morphism of topological spaces $f\colon X\to Y$, we have an inverse image functor
  
  \[ f^{-1}\colon \mathsf{PSh}\webleft (Y\webright )\to \mathsf{PSh}\webleft (X\webright ). \]
  
  The right adjoint of this functor is the direct image functor of presheaves, defined in .
2.
The presheaf direct image functor is denoted $f_{*}$, but the direct image functor for powersets is denoted $f_{!}$ (as it’s a left adjoint).
3.
Adding to the confusion, it’s somewhat common for $f_{!}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright )$ to be denoted $f_{*}$.

We chose to write $f_{!}$ for the direct image to keep the notation aligned with the following similar adjoint situations:

Situation	Adjoint String
Functoriality of Powersets	$\webleft (f_{!}\dashv f^{-1}\dashv f_{*}\webright )\colon \mathcal{P}\webleft (X\webright )\overset {\rightleftarrows }{\to }\mathcal{P}\webleft (Y\webright )$
Functoriality of Presheaf Categories	$\webleft (f_{!}\dashv f^{-1}\dashv f_{*}\webright )\colon \mathsf{PSh}\webleft (X\webright )\overset {\rightleftarrows }{\to }\mathsf{PSh}\webleft (Y\webright )$
Base Change	$\webleft (f_{!}\dashv f^{}\dashv f_{}\webright )\colon \mathcal{C}_{/X}\overset {\rightleftarrows }{\to }\mathcal{C}_{/Y}$
Kan Extensions	$\webleft (F_{!}\dashv F^{}\dashv F_{}\webright )\colon \mathsf{Fun}\webleft (\mathcal{C},\mathcal{E}\webright )\overset {\rightleftarrows }{\to }\mathsf{Fun}\webleft (\mathcal{D},\mathcal{E}\webright )$

ゴ

Remark 4.6.1.1.4 ▶ Unwinding Definition 4.6.1.1.1

Identifying $\mathcal{P}\webleft (X\webright )$ with $\mathsf{Sets}\webleft (X,\{ \mathsf{t},\mathsf{f}\} \webright )$ via Item 2 of Proposition 4.5.1.1.4, we see that the direct image function associated to $f$ is equivalently the function

\[ f_{!}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright ) \]

defined by

\begin{align*} f_{!}\webleft (\chi _{U}\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\operatorname {\mathrm{Lan}}_{f}\webleft (\chi _{U}\webright )\\ & = \operatorname*{\operatorname {\mathrm{colim}}}\webleft (\webleft (f\mathbin {\overset {\to }{\times }}\underline{\webleft (-_{1}\webright )}\webright )\overset {\operatorname {\mathrm{\mathrm{pr}}}}{\twoheadrightarrow }A\overset {\chi _{U}}{\to }\{ \mathsf{t},\mathsf{f}\} \webright )\\ & = \operatorname*{\operatorname {\mathrm{colim}}}_{\substack {x\in X\\ \begin{bgroup} f\webleft (x\webright )=-_{1} \end{bgroup}}}\webleft (\chi _{U}\webleft (x\webright )\webright )\\ & = \bigvee _{\substack {x\in X\\ \begin{bgroup} f\webleft (x\webright )=-_{1} \end{bgroup}}}\webleft (\chi _{U}\webleft (x\webright )\webright ),\end{align*}

where we have used , for the second equality. In other words, we have

\begin{align*} \webleft [f_{!}\webleft (\chi _{U}\webright )\webright ]\webleft (y\webright )& =\bigvee _{\substack {x\in X\\ \begin{bgroup} f\webleft (x\webright )=y \end{bgroup}}}\webleft (\chi _{U}\webleft (x\webright )\webright )\\ & =\begin{cases} \mathsf{true}& \text{if there exists some $x\in X$ such}\\ & \text{that $f\webleft (x\webright )=y$ and $x\in U,$}\\ \mathsf{false}& \text{otherwise} \end{cases}\\ & =\begin{cases} \mathsf{true}& \text{if there exists some $x\in U$}\\ & \text{such that $f\webleft (x\webright )=y$,}\\ \mathsf{false}& \text{otherwise} \end{cases}\end{align*}

for each $y\in Y$.

Proposition 4.6.1.1.5 ▶ Properties of Direct Images I

Let $f\colon X\to Y$ be a function.

1.
Functoriality. The assignment $U\mapsto f_{!}\webleft (U\webright )$ defines a functor

\[ f_{!}\colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (Y\webright ),\subset \webright ). \]

In particular, for each $U,V\in \mathcal{P}\webleft (X\webright )$, the following condition is satisfied:
- (★)
2.
Triple Adjointness. We have a triple adjunction
witnessed by:
1. (a)
  Units and counits of the form
  
  \[ \begin{aligned} \operatorname {\mathrm{id}}_{\mathcal{P}\webleft (X\webright )} & \hookrightarrow f^{-1}\circ f_{!},\\ f_{!}\circ f^{-1} & \hookrightarrow \operatorname {\mathrm{id}}_{\mathcal{P}\webleft (Y\webright )},\\ \end{aligned} \qquad \begin{aligned} \operatorname {\mathrm{id}}_{\mathcal{P}\webleft (Y\webright )} & \hookrightarrow f_{*}\circ f^{-1},\\ f^{-1}\circ f_{*} & \hookrightarrow \operatorname {\mathrm{id}}_{\mathcal{P}\webleft (X\webright )}, \end{aligned} \]
  
  having components of the form
  
  \[ \begin{gathered} U \subset f^{-1}\webleft (f_{!}\webleft (U\webright )\webright ),\\ f_{!}\webleft (f^{-1}\webleft (V\webright )\webright ) \subset V, \end{gathered} \qquad \begin{gathered} V \subset f_{*}\webleft (f^{-1}\webleft (V\webright )\webright ),\\ f^{-1}\webleft (f_{*}\webleft (U\webright )\webright ) \subset U \end{gathered} \]
  
  indexed by $U\in \mathcal{P}\webleft (X\webright )$ and $V\in \mathcal{P}\webleft (Y\webright )$.
2. (b)
  Bijections of sets
  
  \begin{align*} \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (Y\webright )}\webleft (f_{!}\webleft (U\webright ),V\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U,f^{-1}\webleft (V\webright )\webright ),\\ \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (f^{-1}\webleft (U\webright ),V\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U,f_{*}\webleft (V\webright )\webright ), \end{align*}
  
  natural in $U\in \mathcal{P}\webleft (X\webright )$ and $V\in \mathcal{P}\webleft (Y\webright )$ and (respectively) $V\in \mathcal{P}\webleft (X\webright )$ and $U\in \mathcal{P}\webleft (Y\webright )$. In particular:
  1. (i)
    The following conditions are equivalent:
    
    (I)
    We have $f_{!}\webleft (U\webright )\subset V$.
    
    (II)
    We have $U\subset f^{-1}\webleft (V\webright )$.
  2. (ii)
    The following conditions are equivalent:
    
    (I)
    We have $f^{-1}\webleft (U\webright )\subset V$.
    
    (II)
    We have $U\subset f_{*}\webleft (V\webright )$.
3.
Interaction With Unions of Families of Subsets. The diagram
commutes, i.e. we have

\[ \bigcup _{U\in \mathcal{U}}f_{!}\webleft (U\webright )=\bigcup _{V\in f_{!}\webleft (\mathcal{U}\webright )}V \]

for each $\mathcal{U}\in \mathcal{P}\webleft (X\webright )$, where $f_{!}\webleft (\mathcal{U}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{!}\webright )_{!}\webleft (\mathcal{U}\webright )$.
4.
Interaction With Intersections of Families of Subsets. The diagram
commutes, i.e. we have

\[ \bigcap _{U\in \mathcal{U}}f_{!}\webleft (U\webright )=\bigcap _{V\in f_{!}\webleft (\mathcal{U}\webright )}V \]

for each $\mathcal{U}\in \mathcal{P}\webleft (X\webright )$, where $f_{!}\webleft (\mathcal{U}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{!}\webright )_{!}\webleft (\mathcal{U}\webright )$.
5.
Interaction With Binary Unions. The diagram
commutes, i.e. we have

\[ f_{!}\webleft (U\cup V\webright )=f_{!}\webleft (U\webright )\cup f_{!}\webleft (V\webright ) \]

for each $U,V\in \mathcal{P}\webleft (X\webright )$.
6.
Interaction With Binary Intersections. We have a natural transformation
with components

\[ f_{!}\webleft (U\cap V\webright )\subset f_{!}\webleft (U\webright )\cap f_{!}\webleft (V\webright ) \]

indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.
7.
Interaction With Differences. We have a natural transformation
with components

\[ f_{!}\webleft (U\webright )\setminus f_{!}\webleft (V\webright )\subset f_{!}\webleft (U\setminus V\webright ) \]

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

\[ f_{!}\webleft (U^{\textsf{c}}\webright )=f_{*}\webleft (U\webright )^{\textsf{c}} \]

for each $U\in \mathcal{P}\webleft (X\webright )$.
9.
Interaction With Symmetric Differences. We have a natural transformation
with components

\[ f_{!}\webleft (U\webright )\mathbin {\triangle }f_{!}\webleft (V\webright )\subset f_{!}\webleft (U\mathbin {\triangle }V\webright ) \]

indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.
10.
Interaction With Internal Homs of Powersets. The diagram
commutes, i.e. we have an equality of sets

\[ f_{!}\webleft (\webleft [U,V\webright ]_{X}\webright )=\webleft [f_{*}\webleft (U\webright ),f_{!}\webleft (V\webright )\webright ]_{Y}, \]

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
11.
Preservation of Colimits. We have an equality of sets

\[ f_{!}\left(\bigcup _{i\in I}U_{i}\right)=\bigcup _{i\in I}f_{!}\webleft (U_{i}\webright ), \]

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

\[ \begin{gathered} f_{!}\webleft (U\webright )\cup f_{!}\webleft (V\webright ) = f_{!}\webleft (U\cup V\webright ),\\ f_{!}\webleft (\text{Ø}\webright ) = \text{Ø}, \end{gathered} \]

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
12.
Oplax Preservation of Limits. We have an inclusion of sets

\[ f_{!}\left(\bigcap _{i\in I}U_{i}\right)\subset \bigcap _{i\in I}f_{!}\webleft (U_{i}\webright ), \]

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

\[ \begin{gathered} f_{!}\webleft (U\cap V\webright ) \subset f_{!}\webleft (U\webright )\cap f_{!}\webleft (V\webright ),\\ f_{!}\webleft (X\webright ) \subset Y, \end{gathered} \]

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
13.
Symmetric Strict Monoidality With Respect to Unions. The direct image function of Item 1 has a symmetric strict monoidal structure

\[ \webleft (f_{!},f^{\otimes }_{!},f^{\otimes }_{!|\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (X\webright ),\cup ,\text{Ø}\webright ) \to \webleft (\mathcal{P}\webleft (Y\webright ),\cup ,\text{Ø}\webright ), \]

being equipped with equalities

\[ \begin{gathered} f^{\otimes }_{!|U,V} \colon f_{!}\webleft (U\webright )\cup f_{!}\webleft (V\webright ) \mathbin {\overset {=}{\rightarrow }}f_{!}\webleft (U\cup V\webright ),\\ f^{\otimes }_{!|\mathbb {1}} \colon \text{Ø}\mathbin {\overset {=}{\rightarrow }}\text{Ø}, \end{gathered} \]

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
14.
Symmetric Oplax Monoidality With Respect to Intersections. The direct image function of Item 1 has a symmetric oplax monoidal structure

\[ \webleft (f_{!},f^{\otimes }_{!},f^{\otimes }_{!|\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (X\webright ),\cap ,X\webright ) \to \webleft (\mathcal{P}\webleft (Y\webright ),\cap ,Y\webright ), \]

being equipped with inclusions

\[ \begin{gathered} f^{\otimes }_{!|U,V} \colon f_{!}\webleft (U\cap V\webright ) \hookrightarrow f_{!}\webleft (U\webright )\cap f_{!}\webleft (V\webright ),\\ f^{\otimes }_{!|\mathbb {1}} \colon f_{!}\webleft (X\webright ) \hookrightarrow Y, \end{gathered} \]

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
15.
Interaction With Coproducts. Let $f\colon X\to X'$ and $g\colon Y\to Y'$ be maps of sets. We have

\[ \webleft (f\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}g\webright )_{!}\webleft (U\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}V\webright )=f_{!}\webleft (U\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}g_{!}\webleft (V\webright ) \]

for each $U\in \mathcal{P}\webleft (X\webright )$ and each $V\in \mathcal{P}\webleft (Y\webright )$.
16.
Interaction With Products. Let $f\colon X\to X'$ and $g\colon Y\to Y'$ be maps of sets. We have

\[ \webleft (f\boxtimes _{X\times Y} g\webright )_{!}\webleft (U\boxtimes _{X\times Y}V\webright )=f_{!}\webleft (U\webright )\boxtimes _{X'\times Y'}g_{!}\webleft (V\webright ) \]

for each $U\in \mathcal{P}\webleft (X\webright )$ and each $V\in \mathcal{P}\webleft (Y\webright )$.
17.
Relation to Codirect Images. We have

\begin{align*} f_{!}\webleft (U\webright ) & = f_{*}\webleft (U^{\textsf{c}}\webright )^{\textsf{c}}\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}Y\setminus f_{*}\webleft (X\setminus U\webright )\end{align*}

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

Proof of Proposition 4.6.1.1.5.

Item 1: Functoriality

Omitted.

Item 2: Triple Adjointness

This follows from Remark 4.6.1.1.4, Remark 4.6.2.1.2, Remark 4.6.3.1.4, and

Item 3: Interaction With Unions of Families of Subsets

We have

\begin{align*} \bigcup _{V\in f_{!}\webleft (\mathcal{U}\webright )}V & = \bigcup _{V\in \left\{ f_{!}\webleft (U\webright )\in \mathcal{P}\webleft (X\webright )\ \middle |\ U\in \mathcal{U}\right\} }V\\ & = \bigcup _{U\in \mathcal{U}}f_{!}\webleft (U\webright ).\end{align*}

This finishes the proof.

Item 4: Interaction With Intersections of Families of Subsets

We have

\begin{align*} \bigcap _{V\in f_{!}\webleft (\mathcal{U}\webright )}V & = \bigcap _{V\in \left\{ f_{!}\webleft (U\webright )\in \mathcal{P}\webleft (X\webright )\ \middle |\ U\in \mathcal{U}\right\} }V\\ & = \bigcap _{U\in \mathcal{U}}f_{!}\webleft (U\webright ).\end{align*}

This finishes the proof.

Item 5: Interaction With Binary Unions

See [Proof Wiki Contributors, Image of Union Under Mapping — Proof Wiki].

Item 6: Interaction With Binary Intersections

See [Proof Wiki Contributors, Image of Intersection Under Mapping — Proof Wiki].

Item 7: Interaction With Differences

See [Proof Wiki Contributors, Image of Set Difference Under Mapping — Proof Wiki].

Item 8: Interaction With Complements

Applying Item 17 to $X\setminus U$, we have

\begin{align*} f_{!}\webleft (U^{\textsf{c}}\webright ) & = f_{!}\webleft (X\setminus U\webright )\\ & = Y\setminus f_{*}\webleft (X\setminus \webleft (X\setminus U\webright )\webright )\\ & = Y\setminus f_{*}\webleft (U\webright )\\ & = f_{*}\webleft (U\webright )^{\textsf{c}}. \end{align*}

This finishes the proof.

Item 9: Interaction With Symmetric Differences

We have

\begin{align*} f_{!}\webleft (U\webright )\mathbin {\triangle }f_{!}\webleft (V\webright ) & = \webleft (f_{!}\webleft (U\webright )\cup f_{!}\webleft (V\webright )\webright )\setminus \webleft (f_{!}\webleft (U\webright )\cap f_{!}\webleft (V\webright )\webright )\\ & \subset \webleft (f_{!}\webleft (U\webright )\cup f_{!}\webleft (V\webright )\webright )\setminus \webleft (f_{!}\webleft (U\cap V\webright )\webright )\\ & = \webleft (f_{!}\webleft (U\cup V\webright )\webright )\setminus \webleft (f_{!}\webleft (U\cap V\webright )\webright )\\ & \subset f_{!}\webleft (\webleft (U\cup V\webright )\setminus \webleft (U\cap V\webright )\webright )\\ & = f_{!}\webleft (U\mathbin {\triangle }V\webright ), \end{align*}

where we have used:

1.
Item 2 of Proposition 4.3.12.1.2 for the first equality.
2.
Item 6 of this proposition together with Item 1 of Proposition 4.3.10.1.2 for the first inclusion.
3.
Item 5 for the second equality.
4.
Item 7 for the second inclusion.
5.
Item 2 of Proposition 4.3.12.1.2 for the tchird equality.

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

Item 10: Interaction With Internal Homs of Powersets

We have

\begin{align*} f_{!}\webleft (\webleft [U,V\webright ]_{X}\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}\webleft (U^{\textsf{c}}\cup V\webright )\\ & = f_{!}\webleft (U^{\textsf{c}}\webright )\cup f_{!}\webleft (V\webright )\\ & = f_{*}\webleft (U\webright )^{\textsf{c}}\cup f_{!}\webleft (V\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [f_{*}\webleft (U\webright ),f_{!}\webleft (V\webright )\webright ]_{Y},\end{align*}

where we have used:

1.
Item 5 for the second equality.
2.
Item 17 for the third equality.

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

Item 11: Preservation of Colimits

This follows from Item 2 and

.¹

Item 12: Oplax Preservation of Limits

The inclusion $f_{!}\webleft (X\webright )\subset Y$ is automatic. See [Proof Wiki Contributors, Image of Intersection Under Mapping — Proof Wiki] for the other inclusions.

Item 13: Symmetric Strict Monoidality With Respect to Unions

This follows from Item 11.

Item 14: Symmetric Oplax Monoidality With Respect to Intersections

The inclusions in the statement follow from Item 12. Since $\mathcal{P}\webleft (Y\webright )$ is posetal, the commutativity of the diagrams in the definition of a symmetric oplax monoidal functor is automatic (Chapter 11: Categories, Item 4 of Proposition 11.2.7.1.2).

Item 15: Interaction With Coproducts

Omitted.

Item 16: Interaction With Products

Omitted.

Item 17: Relation to Codirect Images

Applying Item 16 of Proposition 4.6.3.1.7 to $X\setminus U$, we have

\begin{align*} f_{*}\webleft (X\setminus U\webright ) & = B\setminus f_{!}\webleft (X\setminus \webleft (X\setminus U\webright )\webright )\\ & = B\setminus f_{!}\webleft (U\webright ). \end{align*}

Taking complements, we then obtain

\begin{align*} f_{!}\webleft (U\webright ) & = B\setminus \webleft (B\setminus f_{!}\webleft (U\webright )\webright ),\\ & = B\setminus f_{*}\webleft (X\setminus U\webright ), \end{align*}

which finishes the proof.

¹Reference: [Proof Wiki Contributors, Image of Union Under Mapping — Proof Wiki].

Proposition 4.6.1.1.6 ▶ Properties of Direct Images II

Let $f\colon X\to Y$ be a function.

1.
Functionality I. The assignment $f\mapsto f_{!}$ defines a function

\[ \webleft (-\webright )_{*|X,Y}\colon \mathsf{Sets}\webleft (X,Y\webright ) \to \mathsf{Sets}\webleft (\mathcal{P}\webleft (X\webright ),\mathcal{P}\webleft (Y\webright )\webright ). \]
2.
Functionality II. The assignment $f\mapsto f_{!}$ defines a function

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

\[ \webleft (\operatorname {\mathrm{id}}_{X}\webright )_{!}=\operatorname {\mathrm{id}}_{\mathcal{P}\webleft (X\webright )}. \]
4.
Interaction With Composition. For each pair of composable functions $f\colon X\to Y$ and $g\colon Y\to Z$, we have