6.4.4 Direct Images With Compact Support

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

Definition 6.4.4.1.1 ▶ Direct Images With Compact Support

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

\[ R_{!}\colon \mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright ) \]

defined by[1][2]

\begin{align*} R_{!}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, if we have}\\ & \text{$b\in R\webleft (a\webright )$, then $a\in U$}\end{aligned} \webright\} \\ & = \webleft\{ b\in B\ \middle |\ R^{-1}\webleft (b\webright )\subset U\webright\} \end{align*}

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

Remark 6.4.4.1.2 ▶ Unwinding Definition 6.4.4.1.1

Identifying subsets of $B$ with relations from $\text{pt}$ to $B$ via Chapter 2: Constructions With Sets, Item 3 of Proposition 2.4.3.1.6, we see that the direct image with compact support function associated to $R$ is equivalently the function

\[ R_{!}\colon \underbrace{\mathcal{P}\webleft (A\webright )}_{\cong \mathrm{Rel}\webleft (A,\text{pt}\webright )}\to \underbrace{\mathcal{P}\webleft (B\webright )}_{\cong \mathrm{Rel}\webleft (B,\text{pt}\webright )} \]

defined by

being explicitly computed by

\begin{align*} R^{*}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\text{Ran}_{R}\webleft (U\webright )\\ & \cong \int _{a\in A}\textup{Hom}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{-_{2}}_{a},U^{-_{1}}_{a}\webright ), \end{align*}

where we have used Proposition 6.2.3.1.1.

Proof of Remark 6.4.4.1.2.

We have

\begin{align*} \text{Ran}_{R}\webleft (V\webright )& \cong \int _{a\in A}\textup{Hom}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{-_{2}}_{a},U^{-_{1}}_{a}\webright )\\ & =\webleft\{ b\in B\ \middle |\ \int _{a\in A}\textup{Hom}_{\{ \mathsf{t},\mathsf{f}\} }\webleft (R^{b}_{a},U^{\star }_{a}\webright )=\mathsf{true}\webright\} \\ & = \webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, at least one of the}\\[-2.5pt]& \text{following conditions hold:}\\[7.5pt]& \mspace {25mu}\rlap {\text{1.}}\mspace {22.5mu}\text{We have $R^{b}_{a}=\mathsf{false}$}\\ & \mspace {25mu}\rlap {\text{2.}}\mspace {22.5mu}\text{The following conditions hold:}\\[7.5pt]& \mspace {50mu}\rlap {\text{(a)}}\mspace {30mu}\text{We have $R^{b}_{a}=\mathsf{true}$}\\ & \mspace {50mu}\rlap {\text{(b)}}\mspace {30mu}\text{We have $U^{\star }_{a}=\mathsf{true}$}\\[10pt]\end{aligned} \webright\} \\ & = \webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, at least one of the}\\[-2.5pt]& \text{following conditions hold:}\\[7.5pt]& \mspace {25mu}\rlap {\text{1.}}\mspace {22.5mu}\text{We have $b\not\in R\webleft (A\webright )$}\\ & \mspace {25mu}\rlap {\text{2.}}\mspace {22.5mu}\text{The following conditions hold:}\\[7.5pt]& \mspace {50mu}\rlap {\text{(a)}}\mspace {30mu}\text{We have $b\in R\webleft (a\webright )$}\\ & \mspace {50mu}\rlap {\text{(b)}}\mspace {30mu}\text{We have $a\in U$}\\[10pt]\end{aligned} \webright\} \\ & = \webleft\{ b\in B\ \middle |\ \begin{aligned} & \text{for each $a\in A$, if we have}\\ & \text{$b\in R\webleft (a\webright )$, then $a\in U$}\end{aligned} \webright\} \\ & = \webleft\{ b\in B\ \middle |\ R^{-1}\webleft (b\webright )\subset U\webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R^{-1}\webleft (U\webright ).\end{align*}

This finishes the proof.

Proposition 6.4.4.1.3 ▶ Properties of Direct Images With Compact Support

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

  1. 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 $R_{!}\webleft (U\webright )\subset R_{!}\webleft (V\webright )$.

  2. 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 $U\subset R_{!}\webleft (V\webright )$.

  3. 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

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

    natural in $U,V\in \mathcal{P}\webleft (A\webright )$.

  4. 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 $U,V\in \mathcal{P}\webleft (A\webright )$.

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

    being equipped with inclusions

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

    natural in $U,V\in \mathcal{P}\webleft (A\webright )$.

  6. 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 $U,V\in \mathcal{P}\webleft (A\webright )$.

  7. Relation to Direct Images. We have
    \[ R_{!}\webleft (U\webright )=B\setminus R_{*}\webleft (A\setminus U\webright ) \]

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

Proof of Proposition 6.4.4.1.3.
Item 1: Functoriality
Clear.
Item 2: Adjointness
This follows from of .
Item 3: Lax Preservation of Colimits
Omitted.
Item 4: Preservation of Limits
This follows from Item 2 and of .
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 6.4.1.1.3. 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 9 of Proposition 2.4.6.1.6). 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.

Proposition 6.4.4.1.4 ▶ Properties of the Direct Image With Compact Support Function Operation

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

  1. Functionality I. The assignment $R\mapsto R_{!}$ defines a function
    \[ \webleft (-\webright )_{!}\colon \mathsf{Sets}\webleft (A,B\webright ) \to \mathsf{Sets}\webleft (\mathcal{P}\webleft (A\webright ),\mathcal{P}\webleft (B\webright )\webright ). \]
  2. Functionality II. The assignment $R\mapsto R_{!}$ defines a function
    \[ \webleft (-\webright )_{!}\colon \mathsf{Sets}\webleft (A,B\webright ) \to \textup{Hom}_{\mathsf{Pos}}\webleft (\webleft (\mathcal{P}\webleft (A\webright ),\subset \webright ),\webleft (\mathcal{P}\webleft (B\webright ),\subset \webright )\webright ). \]
  3. Interaction With Identities. For each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
    \[ \webleft (\text{id}_{A}\webright )_{!}=\text{id}_{\mathcal{P}\webleft (A\webright )}. \]
  4. Interaction With Composition. For each pair of composable relations $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ and $S\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}C$, we have

Proof of Proposition 6.4.4.1.4.
Item 1: Functionality I
Clear.
Item 2: Functionality II
Clear.
Item 3: Interaction With Identities
Indeed, we have
\begin{align*} \webleft (\chi _{A}\webright )_{!}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ a\in A\ \middle |\ \chi ^{-1}_{A}\webleft (a\webright )\subset U\webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ a\in A\ \middle |\ \webleft\{ a\webright\} \subset U\webright\} \\ & = U \end{align*}

for each $U\in \mathcal{P}\webleft (A\webright )$. Thus $\webleft (\chi _{A}\webright )_{!}=\text{id}_{\mathcal{P}\webleft (A\webright )}$.

Item 4: Interaction With Composition
Indeed, we have
\begin{align*} \webleft (S\mathbin {\diamond }R\webright )_{!}\webleft (U\webright ) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ c\in C\ \middle |\ \webleft [S\mathbin {\diamond }R\webright ]^{-1}\webleft (c\webright )\subset U\webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ c\in C\ \middle |\ S^{-1}\webleft (R^{-1}\webleft (c\webright )\webright )\subset U\webright\} \\ & = \webleft\{ c\in C\ \middle |\ R^{-1}\webleft (c\webright )\subset S_{!}\webleft (U\webright )\webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{!}\webleft (S_{!}\webleft (U\webright )\webright )\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft [R_{!}\circ S_{!}\webright ]\webleft (U\webright ) \end{align*}

for each $U\in \mathcal{P}\webleft (C\webright )$, where we used Item 2 of Proposition 6.4.4.1.3, which implies that the conditions

  • We have $S^{-1}\webleft (R^{-1}\webleft (c\webright )\webright )\subset U$.
  • We have $R^{-1}\webleft (c\webright )\subset S_{!}\webleft (U\webright )$.
are equivalent. Thus $\webleft (S\mathbin {\diamond }R\webright )_{!}=S_{!}\circ R_{!}$.

Footnotes

[1] Further Terminology: The set $R_{!}\webleft (U\webright )$ is called the direct image with compact support of $U$ by $R$.
[2] We also have
\[ R_{!}\webleft (U\webright )=B\setminus R_{*}\webleft (A\setminus U\webright ); \]
see Item 7 of Proposition 6.4.4.1.3.
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