Proposition 7.4.4.1.3 (00SZ): Properties of Direct Images With Compact Support

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

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

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

natural in $U\in \mathcal{P}\webleft (A\webright )$ and $V\in \mathcal{P}\webleft (B\webright )$, i.e. such that:
- The following conditions are equivalent:
  - We have $R^{-1}\webleft (U\webright )\subset V$;
  - We have $U\subset R_{!}\webleft (V\webright )$.
Lax Preservation of Colimits. We have an inclusion of sets
\[ \bigcup _{i\in I}R_{!}\webleft (U_{i}\webright )\subset R_{!}\webleft (\bigcup _{i\in I}U_{i}\webright ), \]

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

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

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

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

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

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

being equipped with inclusions

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

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

being equipped with equalities

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

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

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

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: Relation to Direct Images

This follows from Item 7 of Proposition 7.4.1.1.3. Alternatively, we may prove it directly as follows, with the proof proceeding in the same way as in the case of functions (Chapter 2: Constructions With Sets, Item 16 of Proposition 2.6.3.1.6). We claim that $R_{!}\webleft (U\webright )=B\setminus R_{*}\webleft (A\setminus U\webright )$:

The First Implication. We claim that

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

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

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

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

This finishes the proof.