The Clowder Project

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

1. 1.

   Functoriality. The assignment $V\mapsto f^{-1}(V)$ defines a functor

   \[ f^{-1}\colon (\mathcal{P}(Y),\subset )\to (\mathcal{P}(X),\subset ). \]

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

   • (★)
   If $U\subset V$, then $f^{-1}(U)\subset f^{-1}(V)$.
2. 2.

   Triple Adjointness. We have a triple adjunction

   
   witnessed by:
   1. (a)

      Units and counits of the form

      \[ \begin{aligned} \operatorname {\mathrm{id}}_{\mathcal{P}(X)} & \hookrightarrow f^{-1}\circ f_{!},\\ f_{!}\circ f^{-1} & \hookrightarrow \operatorname {\mathrm{id}}_{\mathcal{P}(Y)},\\ \end{aligned} \qquad \begin{aligned} \operatorname {\mathrm{id}}_{\mathcal{P}(Y)} & \hookrightarrow f_{*}\circ f^{-1},\\ f^{-1}\circ f_{*} & \hookrightarrow \operatorname {\mathrm{id}}_{\mathcal{P}(X)}. \end{aligned} \]

      In particular:

      • •

        For each $U\in \mathcal{P}(X)$, we have $U\subset f^{-1}(f_{!}(U))$.

      • •

        For each $U\in \mathcal{P}(X)$, we have $f^{-1}(f_{*}(U))\subset U$.

      • •

        For each $V\in \mathcal{P}(Y)$, we have $f_{!}(f^{-1}(V))\subset V$.

      • •

        For each $V\in \mathcal{P}(Y)$, we have $V\subset f_{*}(f^{-1}(V))$.

   2. (b)

      Bijections of sets

      \begin{align*} \operatorname {\mathrm{Hom}}_{\mathcal{P}(Y)}(f_{!}(U),V) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(U,f^{-1}(V)),\\ \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(f^{-1}(U),V) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(U,f_{*}(V)), \end{align*}

      natural in $U\in \mathcal{P}(X)$ and $V\in \mathcal{P}(Y)$ and (respectively) $V\in \mathcal{P}(X)$ and $U\in \mathcal{P}(Y)$. In particular:

      1. (i)

         The following conditions are equivalent:

         1. (I)

            We have $f_{!}(U)\subset V$.

         2. (II)

            We have $U\subset f^{-1}(V)$.

      2. (ii)

         The following conditions are equivalent:

         1. (I)

            We have $f^{-1}(U)\subset V$.

         2. (II)

            We have $U\subset f_{*}(V)$.

3. 3.

   Interaction With Unions of Families of Subsets. The diagram

   
   commutes, i.e. we have
   \[ \bigcup _{V\in \mathcal{V}}f^{-1}(V)=\bigcup _{U\in f^{-1}(\mathcal{U})}U \]

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

4. 4.

   Interaction With Intersections of Families of Subsets. The diagram

   
   commutes, i.e. we have
   \[ \bigcap _{V\in \mathcal{V}}f^{-1}(V)=\bigcap _{U\in f^{-1}(\mathcal{U})}U \]

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

5. 5.

   Interaction With Binary Unions. The diagram

   
   commutes, i.e. we have
   \[ f^{-1}(U\cup V)=f^{-1}(U)\cup f^{-1}(V) \]

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

6. 6.

   Interaction With Binary Intersections. The diagram

   
   commutes, i.e. we have
   \[ f^{-1}(U\cap V)=f^{-1}(U)\cap f^{-1}(V) \]

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

7. 7.

   Interaction With Differences. The diagram

   
   commutes, i.e. we have
   \[ f^{-1}(U\setminus V)=f^{-1}(U)\setminus f^{-1}(V) \]

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

8. 8.

   Interaction With Complements. The diagram

   
   commutes, i.e. we have
   \[ f^{-1}(U^{\textsf{c}})=f^{-1}(U)^{\textsf{c}} \]

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

9. 9.

   Interaction With Symmetric Differences. The diagram

   
   i.e. we have
   \[ f^{-1}(U)\mathbin {\triangle }f^{-1}(V)=f^{-1}(U\mathbin {\triangle }V) \]

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

10. 10.

    Interaction With Internal Homs of Powersets. The diagram

    
    commutes, i.e. we have an equality of sets
    \[ f^{-1}([U,V]_{Y})=[f^{-1}(U),f^{-1}(V)]_{X}, \]

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

11. 11.

    Preservation of Colimits. We have an equality of sets

    \[ f^{-1}\left(\bigcup _{i\in I}U_{i}\right)=\bigcup _{i\in I}f^{-1}(U_{i}), \]

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

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

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

12. 12.

    Preservation of Limits. We have an equality of sets

    \[ f^{-1}\left(\bigcap _{i\in I}U_{i}\right)=\bigcap _{i\in I}f^{-1}(U_{i}), \]

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

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

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

13. 13.

    Symmetric Strict Monoidality With Respect to Unions. The inverse image function of Item 1 has a symmetric strict monoidal structure

    \[ (f^{-1},f^{-1,\otimes },f^{-1,\otimes }_{\mathbb {1}}) \colon (\mathcal{P}(Y),\cup ,\text{Ø}) \to (\mathcal{P}(X),\cup ,\text{Ø}), \]

    being equipped with equalities

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

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

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

1. 1.

   Functionality I. The assignment $f\mapsto f^{-1}$ defines a function

   \[ (-)^{-1}_{X,Y}\colon \mathsf{Sets}(X,Y) \to \mathsf{Sets}(\mathcal{P}(Y),\mathcal{P}(X)). \]
2. 2.

   Functionality II. The assignment $f\mapsto f^{-1}$ defines a function

   \[ (-)^{-1}_{X,Y}\colon \mathsf{Sets}(X,Y) \to \mathsf{Pos}((\mathcal{P}(Y),\subset ),(\mathcal{P}(X),\subset )). \]
3. 3.

   Interaction With Identities. For each $X\in \operatorname {\mathrm{Obj}}(\mathsf{Sets})$, we have

   \[ \operatorname {\mathrm{id}}^{-1}_{X}=\operatorname {\mathrm{id}}_{\mathcal{P}(X)}. \]
4. 4.

   Interaction With Composition. For each pair of composable functions $f\colon X\to Y$ and $g\colon Y\to Z$, we have