Subsection 2.6.3 (008H): Direct Images With Compact Support

Style	Class	Font	Theorem Environments
Style 1	`book`	Alegreya Sans	`tcbthm`
Style 2	`book`	Alegreya Sans	`amsthm`
Style 3	`book`	Arno*	`amsthm`
Style 4	`book`	Computer Modern	`amsthm`

2.6.3 Direct Images With Compact Support

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

Definition 2.6.3.1.1 ▶ Direct Images With Compact Support

The direct image with compact support function associated to $f$ is the function

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

defined by¹ ²

\begin{align*} f_{!}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ y\in Y\ \middle |\ \begin{aligned} & \text{for each $x\in X$, if we have}\\ & \text{$f\webleft (x\webright )=y$, then $x\in U$}\end{aligned} \webright\} \\ & = \webleft\{ y\in Y\ \middle |\ \text{we have $f^{-1}\webleft (y\webright )\subset U$}\webright\} \end{align*}

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

¹Further Terminology: The set $f_{!}\webleft (U\webright )$ is called the direct image with compact support of $U$ by $f$.

²We also 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*}

see Item 16 of Proposition 2.6.3.1.6.

Notation 2.6.3.1.2 ▶ Further Notation for Direct Images With Compact Support

Sometimes one finds the notation

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

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 $y\in \forall _{f}\webleft (U\webright )$.
For each $x\in X$, if $y=f\webleft (x\webright )$, then $x\in U$.

We will not make use of this notation in the present work.

Remark 2.6.3.1.3 ▶ Unwinding Definition 2.6.3.1.1

Identifying $\mathcal{P}\webleft (X\webright )$ with $\mathsf{Sets}\webleft (X,\{ \mathsf{t},\mathsf{f}\} \webright )$ via Item 2 of Proposition 2.5.1.1.4, we see that the direct image with compact support 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}}}=}}\text{Ran}_{f}\webleft (\chi _{U}\webright )\\ & = \operatorname*{\text{lim}}\webleft (\webleft (\underline{\webleft (-_{1}\webright )}\mathbin {\overset {\to }{\times }}f\webright )\overset {\text{pr}}{\twoheadrightarrow }X\overset {\chi _{U}}{\to }\{ \mathsf{true},\mathsf{false}\} \webright )\\ & = \operatorname*{\text{lim}}_{\substack {x\in X\\ f\webleft (x\webright )=-_{1} }}\webleft (\chi _{U}\webleft (x\webright )\webright )\\ & = \bigwedge _{\substack {x\in X\\ f\webleft (x\webright )=-_{1} }}\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 ) & = \bigwedge _{\substack {x\in X\\ f\webleft (x\webright )=y }}\webleft (\chi _{U}\webleft (x\webright )\webright )\\ & = \begin{cases} \mathsf{true}& \text{if, for each $x\in X$ such that}\\ & \text{$f\webleft (x\webright )=y$, we have $x\in U,$}\\ \mathsf{false}& \text{otherwise} \end{cases}\\ & = \begin{cases} \mathsf{true}& \text{if $f^{-1}\webleft (y\webright )\subset U$}\\ \mathsf{false}& \text{otherwise} \end{cases}\end{align*}

for each $y\in Y$.

Definition 2.6.3.1.4 ▶ The Image and Complement Parts of $f_{!}$

Let $U$ be a subset of $X$.¹ ²

The image part of the direct image with compact support $f_{!}\webleft (U\webright )$ of $U$ is the set $f_{!,\text{im}}\webleft (U\webright )$ defined by
\begin{align*} f_{!,\text{im}}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}\webleft (U\webright )\cap \mathrm{Im}\webleft (f\webright )\\ & = \webleft\{ y\in Y\ \middle |\ \text{we have $f^{-1}\webleft (y\webright )\subset U$ and $f^{-1}\webleft (y\webright )\neq \text{Ø}$}\webright\} .\end{align*}
The complement part of the direct image with compact support $f_{!}\webleft (U\webright )$ of $U$ is the set $f_{!,\text{cp}}\webleft (U\webright )$ defined by
\begin{align*} f_{!,\text{cp}}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}\webleft (U\webright )\cap \webleft (Y\setminus \mathrm{Im}\webleft (f\webright )\webright )\\ & = Y\setminus \mathrm{Im}\webleft (f\webright )\\ & = \webleft\{ y\in Y\ \middle |\ \text{we have $f^{-1}\webleft (y\webright )\subset U$ and $f^{-1}\webleft (y\webright )=\text{Ø}$}\webright\} \\ & = \webleft\{ y\in Y\ \middle |\ f^{-1}\webleft (y\webright )=\text{Ø}\webright\} .\end{align*}

¹Note that we have

\[ f_{!}\webleft (U\webright )=f_{!,\text{im}}\webleft (U\webright )\cup f_{!,\text{cp}}\webleft (U\webright ), \]

\begin{align*} f_{!}\webleft (U\webright ) & = f_{!}\webleft (U\webright )\cap Y\\ & = f_{!}\webleft (U\webright )\cap \webleft (\mathrm{Im}\webleft (f\webright )\cup \webleft (Y\setminus \mathrm{Im}\webleft (f\webright )\webright )\webright )\\ & = \webleft (f_{!}\webleft (U\webright )\cap \mathrm{Im}\webleft (f\webright )\webright )\cup \webleft (f_{!}\webleft (U\webright )\cap \webleft (Y\setminus \mathrm{Im}\webleft (f\webright )\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!,\text{im}}\webleft (U\webright )\cup f_{!,\text{cp}}\webleft (U\webright ). \end{align*}

²In terms of the meet computation of $f_{!}\webleft (U\webright )$ of Remark 2.6.3.1.3, namely

\[ f_{!}\webleft (\chi _{U}\webright ) =\bigwedge _{\substack {x\in X\\ f\webleft (x\webright )=-_{1}}}\webleft (\chi _{U}\webleft (x\webright )\webright ), \]

we see that $\smash {f_{!,\text{im}}}$ corresponds to meets indexed over nonempty sets, while $\smash {f_{!,\text{cp}}}$ corresponds to meets indexed over the empty set.

Example 2.6.3.1.5 ▶ Examples of Direct Images With Compact Support

Here are some examples of direct images with compact support.

The Multiplication by Two Map on the Natural Numbers. Consider the function $f\colon \mathbb {N}\to \mathbb {N}$ given by
\[ f\webleft (n\webright ) \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}2n \]

for each $n\in \mathbb {N}$. Since $f$ is injective, we have

\begin{align*} f_{!,\text{im}}\webleft (U\webright ) & = f_{*}\webleft (U\webright )\\ f_{!,\text{cp}}\webleft (U\webright ) & = \webleft\{ \text{odd natural numbers}\webright\} \end{align*}

for any $U\subset \mathbb {N}$.
Parabolas. Consider the function $f\colon \mathbb {R}\to \mathbb {R}$ given by
\[ f\webleft (x\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x^{2} \]

for each $x\in \mathbb {R}$. We have

\[ f_{!,\text{cp}}\webleft (U\webright )=\mathbb {R}_{<0} \]

for any $U\subset \mathbb {R}$. Moreover, since $f^{-1}\webleft (x\webright )=\webleft\{ -\sqrt{x},\sqrt{x}\webright\} $, we have e.g.:

\begin{gather*} f_{!,\text{im}}\webleft (\webleft [0,1\webright ]\webright ) = \webleft\{ 0\webright\} ,\\ f_{!,\text{im}}\webleft (\webleft [-1,1\webright ]\webright ) = \webleft [0,1\webright ],\\ f_{!,\text{im}}\webleft (\webleft [1,2\webright ]\webright ) = \text{Ø},\\ f_{!,\text{im}}\webleft (\webleft [-2,-1\webright ]\cup \webleft [1,2\webright ]\webright ) = \webleft [1,4\webright ]. \end{gather*}
Circles. Consider the function $f\colon \mathbb {R}^{2}\to \mathbb {R}$ given by
\[ f\webleft (x,y\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}x^{2}+y^{2} \]

for each $\webleft (x,y\webright )\in \mathbb {R}^{2}$. We have

\[ f_{!,\text{cp}}\webleft (U\webright )=\mathbb {R}_{<0} \]

for any $U\subset \mathbb {R}^{2}$, and since

\[ f^{-1}\webleft (r\webright )= \begin{cases} \text{a circle of radius $r$ about the origin} & \text{if $r>0$,}\\ \webleft\{ \webleft (0,0\webright )\webright\} & \text{if $r=0$,}\\ \text{Ø}& \text{if $r<0$,} \end{cases} \]

we have e.g.:

\begin{gather*} f_{!,\text{im}}\webleft (\webleft [-1,1\webright ]\times \webleft [-1,1\webright ]\webright ) = \webleft [0,1\webright ],\\ f_{!,\text{im}}\webleft (\webleft (\webleft [-1,1\webright ]\times \webleft [-1,1\webright ]\webright )\setminus \webleft [-1,1\webright ]\times \webleft\{ 0\webright\} \webright ) = \text{Ø}. \end{gather*}

Proposition 2.6.3.1.6 ▶ Properties of Direct Images With Compact Support I

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

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:
- If $U\subset V$, then $f_{!}\webleft (U\webright )\subset f_{!}\webleft (V\webright )$.
Triple Adjointness. We have a triple adjunction
witnessed by:
1. Units and counits of the form
  \[ \begin{aligned} \text{id}_{\mathcal{P}\webleft (X\webright )} & \hookrightarrow f^{-1}\circ f_{*},\\ f_{*}\circ f^{-1} & \hookrightarrow \text{id}_{\mathcal{P}\webleft (Y\webright )},\\ \end{aligned} \qquad \begin{aligned} \text{id}_{\mathcal{P}\webleft (Y\webright )} & \hookrightarrow f_{!}\circ f^{-1},\\ f^{-1}\circ f_{!} & \hookrightarrow \text{id}_{\mathcal{P}\webleft (X\webright )}, \end{aligned} \]
  
  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. Bijections of sets
  \begin{align*} \textup{Hom}_{\mathcal{P}\webleft (Y\webright )}\webleft (f_{*}\webleft (U\webright ),V\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,f^{-1}\webleft (V\webright )\webright ),\\ \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (f^{-1}\webleft (U\webright ),V\webright ) & \cong \textup{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. The following conditions are equivalent:
    1. We have $f_{*}\webleft (U\webright )\subset V$.
    2. We have $U\subset f^{-1}\webleft (V\webright )$.
  2. The following conditions are equivalent:
    1. We have $f^{-1}\webleft (U\webright )\subset V$.
    2. We have $U\subset f_{!}\webleft (V\webright )$.
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 $f_{!}\webleft (\mathcal{U}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{!}\webright )_{!}\webleft (\mathcal{U}\webright )$.
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 $f_{!}\webleft (\mathcal{U}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{!}\webright )_{!}\webleft (\mathcal{U}\webright )$.
Interaction With Binary Unions. Let $f\colon X\to Y$ be a function. We have a natural transformation

with components

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

indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.
Interaction With Binary Intersections. The diagram

commutes, i.e. we have

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

for each $U,V\in \mathcal{P}\webleft (X\webright )$.
Interaction With Complements. The diagram

commutes, i.e. we have

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

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

with components

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

indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.
Interaction With Internal Homs of Powersets. We have a natural transformation

with components

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

indexed by $U,V\in \mathcal{P}\webleft (X\webright )$.
Lax Preservation of Colimits. We have an inclusion of sets
\[ \bigcup _{i\in I}f_{!}\webleft (U_{i}\webright )\subset f_{!}\webleft(\bigcup _{i\in I}U_{i}\webright), \]

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

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

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
Preservation of Limits. We have an equality of sets
\[ f_{!}\webleft(\bigcap _{i\in I}U_{i}\webright)=\bigcap _{i\in I}f_{!}\webleft (U_{i}\webright ), \]

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

\[ \begin{gathered} f^{-1}\webleft (U\cap V\webright ) = f_{!}\webleft (U\webright )\cap f^{-1}\webleft (V\webright ),\\ f_{!}\webleft (X\webright ) = Y, \end{gathered} \]

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
Symmetric Lax Monoidality With Respect to Unions. The direct image with compact support function of Item 1 has a symmetric lax 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 inclusions

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

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
Symmetric Strict Monoidality With Respect to Intersections. The direct image function of Item 1 has a symmetric strict monoidal structure
\[ \webleft (f_{!},f^{\otimes }_{!},f^{\otimes }_{!|\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (X\webright ),\cap ,X\webright ) \to \webleft (\mathcal{P}\webleft (Y\webright ),\cap ,Y\webright ), \]

being equipped with equalities

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

natural in $U,V\in \mathcal{P}\webleft (X\webright )$.
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 $V\in \mathcal{P}\webleft (Y\webright )$.
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 $V\in \mathcal{P}\webleft (Y\webright )$.
Relation to Direct Images. We have
\begin{align*} f_{!}\webleft (U\webright ) & = f_{*}\webleft (U^{\textsf{c}}\webright )^{\textsf{c}}\\ & = Y\setminus f_{*}\webleft (X\setminus U\webright ) \end{align*}

for each $U\in \mathcal{P}\webleft (X\webright )$.
Interaction With Injections. If $f$ is injective, then we have
\begin{gather*} f_{!,\text{im}}\webleft (U\webright ) = f_{*}\webleft (U\webright ),\\ f_{!,\text{cp}}\webleft (U\webright ) = Y\setminus \mathrm{Im}\webleft (f\webright ),\\ \begin{aligned} f_{!}\webleft (U\webright ) & = f_{!,\text{im}}\webleft (U\webright )\cup f_{!,\text{cp}}\webleft (U\webright )\\ & = f_{*}\webleft (U\webright )\cup \webleft (Y\setminus \mathrm{Im}\webleft (f\webright )\webright ) \end{aligned}\end{gather*}

for each $U\in \mathcal{P}\webleft (X\webright )$.
Interaction With Surjections. If $f$ is surjective, then we have
\begin{align*} f_{!,\text{im}}\webleft (U\webright ) & \subset f_{*}\webleft (U\webright ),\\ f_{!,\text{cp}}\webleft (U\webright ) & = \text{Ø},\\ f_{!}\webleft (U\webright ) & \subset f_{*}\webleft (U\webright ) \end{align*}

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

Proof of Proposition 2.6.3.1.6.

Item 1: Functoriality

Omitted.

Item 2: Triple Adjointness

This follows from Remark 2.6.1.1.3, Remark 2.6.2.1.2, Remark 2.6.3.1.3, 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 \webleft\{ f_{!}\webleft (U\webright )\in \mathcal{P}\webleft (X\webright )\ \middle |\ U\in \mathcal{U}\webright\} }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 \webleft\{ f_{!}\webleft (U\webright )\in \mathcal{P}\webleft (X\webright )\ \middle |\ U\in \mathcal{U}\webright\} }V\\ & = \bigcap _{U\in \mathcal{U}}f_{!}\webleft (U\webright ).\end{align*}

This finishes the proof.

Item 5: Interaction With Binary Unions

We have

\begin{align*} f_{!}\webleft (U\webright )\cup f_{!}\webleft (V\webright ) & = f_{*}\webleft (U^{\textsf{c}}\webright )^{\textsf{c}}\cup f_{*}\webleft (V^{\textsf{c}}\webright )^{\textsf{c}}\\ & = \webleft (f_{*}\webleft (U^{\textsf{c}}\webright )\cap f_{*}\webleft (V^{\textsf{c}}\webright )\webright )^{\textsf{c}}\\ & \subset \webleft (f_{*}\webleft (U^{\textsf{c}}\cap V^{\textsf{c}}\webright )\webright )^{\textsf{c}}\\ & = f_{*}\webleft (\webleft (U\cup V\webright )^{\textsf{c}}\webright )^{\textsf{c}}\\ & = f_{!}\webleft (U\cup V\webright ), \end{align*}

where:

We have used Item 16 for the first equality.
We have used Item 2 of Proposition 2.3.11.1.2 for the second equality.
We have used Item 6 of Proposition 2.6.1.1.4 for the third equality.
We have used Item 2 of Proposition 2.3.11.1.2 for the fourth equality.
We have used Item 16 for the last equality.

This finishes the proof.

Item 7: Interaction With Complements

Omitted.

Item 8: Interaction With Symmetric Differences

Omitted.

Item 9: Interaction With Internal Homs of Powersets

We have

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

where we have used:

Item 7 of Proposition 2.6.3.1.6 for the second equality.
Item 5 of Proposition 2.6.3.1.6 for the inclusion.

Since $\mathcal{P}\webleft (X\webright )$ is posetal, naturality is automatic (). This finishes the proof.

Item 6: Interaction With Binary Intersections

This follows from Item 11.

Item 10: Lax Preservation of Colimits

Omitted.

Item 11: Preservation of Limits

This follows from Item 2 and

Item 12: Symmetric Lax Monoidality With Respect to Unions

This follows from Item 10.

Item 13: Symmetric Strict Monoidality With Respect to Intersections

This follows from Item 11.

Item 14: Interaction With Coproducts

Omitted.

Item 15: Interaction With Products

Omitted.

Item 16: Relation to Direct Images

We claim that $f_{!}\webleft (U\webright )=Y\setminus f_{*}\webleft (X\setminus U\webright )$.

The First Implication. We claim that

\[ f_{!}\webleft (U\webright )\subset Y\setminus f_{*}\webleft (X\setminus U\webright ). \]

Let $y\in f_{!}\webleft (U\webright )$. We need to show that $y\not\in f_{*}\webleft (X\setminus U\webright )$, i.e. that there is no $x\in X\setminus U$ such that $f\webleft (x\webright )=y$.
This is indeed the case, as otherwise we would have $x\in f^{-1}\webleft (y\webright )$ and $x\not\in U$, contradicting $f^{-1}\webleft (y\webright )\subset U$ (which holds since $y\in f_{!}\webleft (U\webright )$).
Thus $y\in Y\setminus f_{*}\webleft (X\setminus U\webright )$.
The Second Implication. We claim that

\[ Y\setminus f_{*}\webleft (X\setminus U\webright )\subset f_{!}\webleft (U\webright ). \]

Let $y\in Y\setminus f_{*}\webleft (X\setminus U\webright )$. We need to show that $y\in f_{!}\webleft (U\webright )$, i.e. that $f^{-1}\webleft (y\webright )\subset U$.
Since $y\not\in f_{*}\webleft (X\setminus U\webright )$, there exists no $x\in X\setminus U$ such that $y=f\webleft (x\webright )$, and hence $f^{-1}\webleft (y\webright )\subset U$.
Thus $y\in f_{!}\webleft (U\webright )$.

This finishes the proof of Item 16.

Item 17: Interaction With Injections

Omitted.

Item 18: Interaction With Surjections

Omitted.

Proposition 2.6.3.1.7 ▶ Properties of Direct Images With Compact Support II

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

Functionality I. The assignment $f\mapsto f_{!}$ defines a function
\[ \webleft (-\webright )_{!|X,Y}\colon \mathsf{Sets}\webleft (X,Y\webright ) \to \mathsf{Sets}\webleft (\mathcal{P}\webleft (X\webright ),\mathcal{P}\webleft (Y\webright )\webright ). \]
Functionality II. The assignment $f\mapsto f_{!}$ defines a function
\[ \webleft (-\webright )_{!|X,Y}\colon \mathsf{Sets}\webleft (X,Y\webright ) \to \mathsf{Pos}\webleft (\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\webleft (\mathcal{P}\webleft (Y\webright ),\subset \webright )\webright ). \]
Interaction With Identities. For each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
\[ \webleft (\text{id}_{X}\webright )_{!}=\text{id}_{\mathcal{P}\webleft (X\webright )}. \]
Interaction With Composition. For each pair of composable functions $f\colon X\to Y$ and $g\colon Y\to Z$, we have