Proposition 2.4.6.1.6 (008R): Properties of Direct Images With Compact Support I

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

Functoriality. The assignment $U\mapsto f_{!}\webleft (U\webright )$ defines a functor
\[ f_{!}\colon \webleft (\mathcal{P}\webleft (A\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (B\webright ),\subset \webright ) \]

where
- Action on Objects. For each $U\in \mathcal{P}\webleft (A\webright )$, we have
  
  \[ \webleft [f_{!}\webright ]\webleft (U\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}f_{!}\webleft (U\webright ). \]
- Action on Morphisms. For each $U,V\in \mathcal{P}\webleft (A\webright )$:
  - If $U\subset V$, then $f_{!}\webleft (U\webright )\subset f_{!}\webleft (V\webright )$.
Triple Adjointness. We have a triple adjunction
witnessed by bijections of sets
\begin{align*} \textup{Hom}_{\mathcal{P}\webleft (B\webright )}\webleft (f_{*}\webleft (U\webright ),V\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (A\webright )}\webleft (U,f^{-1}\webleft (V\webright )\webright ),\\ \textup{Hom}_{\mathcal{P}\webleft (A\webright )}\webleft (f^{-1}\webleft (U\webright ),V\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (A\webright )}\webleft (U,f_{!}\webleft (V\webright )\webright ), \end{align*}

natural in $U\in \mathcal{P}\webleft (A\webright )$ and $V\in \mathcal{P}\webleft (B\webright )$ and (respectively) $V\in \mathcal{P}\webleft (A\webright )$ and $U\in \mathcal{P}\webleft (B\webright )$, i.e. where:
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 $U\subset f_{!}\webleft (V\webright )$.
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 $\webleft\{ U_{i}\webright\} _{i\in I}\in \mathcal{P}\webleft (A\webright )^{\times I}$. In particular, we have inclusions

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

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

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

natural in $U,V\in \mathcal{P}\webleft (A\webright )$.
Symmetric Lax Monoidality With Respect to Unions. The direct image with compact support function of Item 1 has a symmetric lax monoidal structure
\[ \webleft (f_{!},f^{\otimes }_{!},f^{\otimes }_{!|\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (A\webright ),\cup ,\emptyset \webright ) \to \webleft (\mathcal{P}\webleft (B\webright ),\cup ,\emptyset \webright ), \]

being equipped with inclusions

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

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

being equipped with equalities

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

natural in $U,V\in \mathcal{P}\webleft (A\webright )$.
Interaction With Coproducts. Let $f\colon A\to A'$ and $g\colon B\to B'$ 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 $U\in \mathcal{P}\webleft (A\webright )$ and each $V\in \mathcal{P}\webleft (B\webright )$.
Interaction With Products. Let $f\colon A\to A'$ and $g\colon B\to B'$ be maps of sets. We have
\[ \webleft (f\times g\webright )_{!}\webleft (U\times V\webright )=f_{!}\webleft (U\webright )\times g_{!}\webleft (V\webright ) \]

for each $U\in \mathcal{P}\webleft (A\webright )$ and each $V\in \mathcal{P}\webleft (B\webright )$.
Relation to Direct Images. We have
\[ f_{!}\webleft (U\webright )=B\setminus f_{*}\webleft (A\setminus U\webright ) \]

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

for each $U\in \mathcal{P}\webleft (A\webright )$.
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 ) & = \emptyset ,\\ f_{!}\webleft (U\webright ) & \subset f_{*}\webleft (U\webright ) \end{align*}

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

Item 1: Functoriality

Clear.

Item 2: Triple Adjointness

This follows from Remark 2.4.4.1.3, Remark 2.4.5.1.2, Remark 2.4.6.1.3, and

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 Coproducts

Clear.

Item 8: Interaction With Products

Clear.

Item 9: Relation to Direct Images

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

The First Implication. We claim that
\[ f_{!}\webleft (U\webright )\subset B\setminus f_{*}\webleft (A\setminus U\webright ). \]

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

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

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

This finishes the proof of Item 9.

Item 10: Interaction With Injections

Clear.

Item 11: Interaction With Surjections

Clear.