The Clowder Project

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

1. Functoriality. The assignment $V\mapsto f^{-1}\webleft (V\webright )$ defines a functor
   \[ f^{-1}\colon \webleft (\mathcal{P}\webleft (Y\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \]

   In particular, for each $U,V\in \mathcal{P}\webleft (Y\webright )$, the following condition is satisfied:

   • If $U\subset V$, then $f^{-1}\webleft (U\webright )\subset f^{-1}\webleft (V\webright )$.

2. 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 $U\subset f_{!}\webleft (V\webright )$.
3. Interaction With Unions of Families of Subsets. The diagram
   

   commutes, i.e. we have

   \[ \bigcup _{V\in \mathcal{V}}f^{-1}\webleft (V\webright )=\bigcup _{U\in f^{-1}\webleft (\mathcal{U}\webright )}U \]

   for each $\mathcal{V}\in \mathcal{P}\webleft (Y\webright )$, where $f^{-1}\webleft (\mathcal{V}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f^{-1}\webright )^{-1}\webleft (\mathcal{V}\webright )$.

4. Interaction With Intersections of Families of Subsets. The diagram
   

   commutes, i.e. we have

   \[ \bigcap _{V\in \mathcal{V}}f^{-1}\webleft (V\webright )=\bigcap _{U\in f^{-1}\webleft (\mathcal{U}\webright )}U \]

   for each $\mathcal{V}\in \mathcal{P}\webleft (Y\webright )$, where $f^{-1}\webleft (\mathcal{V}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f^{-1}\webright )^{-1}\webleft (\mathcal{V}\webright )$.

5. Interaction With Binary Unions. The diagram
   

   commutes, i.e. we have

   \[ f^{-1}\webleft (U\cup V\webright )=f^{-1}\webleft (U\webright )\cup f^{-1}\webleft (V\webright ) \]

   for each $U,V\in \mathcal{P}\webleft (Y\webright )$.

6. Interaction With Binary Intersections. The diagram
   

   commutes, i.e. we have

   \[ f^{-1}\webleft (U\cap V\webright )=f^{-1}\webleft (U\webright )\cap f^{-1}\webleft (V\webright ) \]

   for each $U,V\in \mathcal{P}\webleft (Y\webright )$.

7. Interaction With Differences. The diagram
   

   commutes, i.e. we have

   \[ f^{-1}\webleft (U\setminus V\webright )=f^{-1}\webleft (U\webright )\setminus f^{-1}\webleft (V\webright ) \]

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

8. Interaction With Complements. The diagram
   

   commutes, i.e. we have

   \[ f^{-1}\webleft (U^{\textsf{c}}\webright )=f^{-1}\webleft (U\webright )^{\textsf{c}} \]

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

9. Interaction With Symmetric Differences. The diagram
   

   i.e. we have

   \[ f^{-1}\webleft (U\webright )\mathbin {\triangle }f^{-1}\webleft (V\webright )=f^{-1}\webleft (U\mathbin {\triangle }V\webright ) \]

   for each $U,V\in \mathcal{P}\webleft (Y\webright )$.

10. Interaction With Internal Homs of Powersets. The diagram
    

    commutes, i.e. we have an equality of sets

    \[ f^{-1}\webleft (\webleft [U,V\webright ]_{Y}\webright )=\webleft [f^{-1}\webleft (U\webright ),f^{-1}\webleft (V\webright )\webright ]_{X}, \]

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

11. Preservation of Colimits. We have an equality of sets
    \[ f^{-1}\webleft(\bigcup _{i\in I}U_{i}\webright)=\bigcup _{i\in I}f^{-1}\webleft (U_{i}\webright ), \]

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

    \[ \begin{gathered} f^{-1}\webleft (U\webright )\cup f^{-1}\webleft (V\webright ) = f^{-1}\webleft (U\cup V\webright ),\\ f^{-1}\webleft (\text{Ø}\webright ) = \text{Ø}, \end{gathered} \]

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

12. Preservation of Limits. We have an equality of sets
    \[ f^{-1}\webleft(\bigcap _{i\in I}U_{i}\webright)=\bigcap _{i\in I}f^{-1}\webleft (U_{i}\webright ), \]

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

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

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

13. Symmetric Strict Monoidality With Respect to Unions. The inverse image function of Item 1 has a symmetric strict monoidal structure
    \[ \webleft (f^{-1},f^{-1,\otimes },f^{-1,\otimes }_{\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (Y\webright ),\cup ,\text{Ø}\webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\cup ,\text{Ø}\webright ), \]

    being equipped with equalities

    \[ \begin{gathered} f^{-1,\otimes }_{U,V} \colon f^{-1}\webleft (U\webright )\cup f^{-1}\webleft (V\webright ) \mathbin {\overset {=}{\rightarrow }}f^{-1}\webleft (U\cup V\webright ),\\ f^{-1,\otimes }_{\mathbb {1}} \colon \text{Ø}\mathbin {\overset {=}{\rightarrow }}f^{-1}\webleft (\text{Ø}\webright ), \end{gathered} \]

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

14. Symmetric Strict Monoidality With Respect to Intersections. The inverse image function of Item 1 has a symmetric strict monoidal structure
    \[ \webleft (f^{-1},f^{-1,\otimes },f^{-1,\otimes }_{\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (Y\webright ),\cap ,Y\webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\cap ,X\webright ), \]

    being equipped with equalities

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

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

15. Interaction With Coproducts. Let $f\colon X\to X'$ and $g\colon Y\to Y'$ be maps of sets. We have
    \[ \webleft (f\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}g\webright )^{-1}\webleft (U'\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}V'\webright )=f^{-1}\webleft (U'\webright )\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}g^{-1}\webleft (V'\webright ) \]

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

16. Interaction With Products. Let $f\colon X\to X'$ and $g\colon Y\to Y'$ be maps of sets. We have
    \[ \webleft (f\boxtimes _{X'\times Y'}g\webright )^{-1}\webleft (U'\boxtimes _{X'\times Y'} V'\webright )=f^{-1}\webleft (U'\webright )\boxtimes _{X\times Y} g^{-1}\webleft (V'\webright ) \]

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

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

1. Functionality I. The assignment $f\mapsto f^{-1}$ defines a function
   \[ \webleft (-\webright )^{-1}_{X,Y}\colon \mathsf{Sets}\webleft (X,Y\webright ) \to \mathsf{Sets}\webleft (\mathcal{P}\webleft (Y\webright ),\mathcal{P}\webleft (X\webright )\webright ). \]
2. Functionality II. The assignment $f\mapsto f^{-1}$ defines a function
   \[ \webleft (-\webright )^{-1}_{X,Y}\colon \mathsf{Sets}\webleft (X,Y\webright ) \to \mathsf{Pos}\webleft (\webleft (\mathcal{P}\webleft (Y\webright ),\subset \webright ),\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )\webright ). \]
3. Interaction With Identities. For each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
   \[ \text{id}^{-1}_{X}=\text{id}_{\mathcal{P}\webleft (X\webright )}. \]
4. Interaction With Composition. For each pair of composable functions $f\colon X\to Y$ and $g\colon Y\to Z$, we have