The Clowder Project

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

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

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

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

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

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

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

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

7. 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 $V\in \mathcal{P}\webleft (B\webright )$.

8. 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 $V\in \mathcal{P}\webleft (B\webright )$.

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

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

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

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

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

1. Functionality I. The assignment $f\mapsto f_{!}$ defines a function
   \[ \webleft (-\webright )_{!|A,B}\colon \mathsf{Sets}\webleft (A,B\webright ) \to \mathsf{Sets}\webleft (\mathcal{P}\webleft (A\webright ),\mathcal{P}\webleft (B\webright )\webright ). \]
2. Functionality II. The assignment $f\mapsto f_{!}$ defines a function
   \[ \webleft (-\webright )_{!|A,B}\colon \mathsf{Sets}\webleft (A,B\webright ) \to \mathsf{Pos}\webleft (\webleft (\mathcal{P}\webleft (A\webright ),\subset \webright ),\webleft (\mathcal{P}\webleft (B\webright ),\subset \webright )\webright ). \]
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 functions $f\colon A\to B$ and $g\colon B\to C$, we have