The Clowder Project

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

1. 1.

   Functoriality. The assignment $U\mapsto R_{!}(U)$ defines a functor

   \[ R_{!}\colon (\mathcal{P}(X),\subset )\to (\mathcal{P}(Y),\subset ). \]

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

   • (★)
   If $U\subset V$, then $R_{!}(U)\subset R_{!}(V)$.
2. 2.

   Adjointness. We have an adjunction

   
   witnessed by:
   1. (a)

      Units and counits of the form

      \begin{align*} \operatorname {\mathrm{id}}_{\mathcal{P}(X)} \hookrightarrow R_{-1}\circ R_{!},\\ R_{!}\circ R_{-1} \hookrightarrow \operatorname {\mathrm{id}}_{\mathcal{P}(Y)}. \end{align*}

      In particular:

      • •

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

      • •

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

   2. (b)

      A bijections of sets

      \[ \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(R_{!}(U),V)\cong \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(U,R_{-1}(V)), \]

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

      1. (i)

         The following conditions are equivalent:

         1. (I)

            we have $R_{!}(U)\subset V$.

         2. (II)

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

3. 3.

   Interaction With Unions of Families of Subsets. The diagram

   
   commutes, i.e. we have
   \[ \bigcup _{U\in \mathcal{U}}R_{!}(U)=\bigcup _{V\in R_{!}(\mathcal{U})}V \]

   for each $\mathcal{U}\in \mathcal{P}(X)$, where $R_{!}(\mathcal{U})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(R_{!})_{!}(\mathcal{U})$.

4. 4.

   Interaction With Intersections of Families of Subsets. The diagram

   
   commutes, i.e. we have
   \[ \bigcap _{U\in \mathcal{U}}R_{!}(U)=\bigcap _{V\in R_{!}(\mathcal{U})}V \]

   for each $\mathcal{U}\in \mathcal{P}(X)$, where $R_{!}(\mathcal{U})\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}(R_{!})_{!}(\mathcal{U})$.

5. 5.

   Interaction With Binary Unions. The diagram

   
   commutes, i.e. we have
   \[ R_{!}(U\cup V)=R_{!}(U)\cup R_{!}(V) \]

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

6. 6.

   Interaction With Binary Intersections. We have a natural transformation

   
   with components
   \[ R_{!}(U\cap V)\subset R_{!}(U)\cap R_{!}(V) \]

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

7. 7.

   Interaction With Differences. We have a natural transformation

   
   with components
   \[ R_{!}(U)\setminus R_{!}(V)\subset R_{!}(U\setminus V) \]

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

8. 8.

   Interaction With Complements. The diagram

   
   commutes, i.e. we have
   \[ R_{!}(U^{\textsf{c}})=R_{*}(U)^{\textsf{c}} \]

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

9. 9.

   Interaction With Binary Symmetric Differences. We have a natural transformation

   
   with components
   \[ R_{!}(U)\mathbin {\triangle }R_{!}(V)\subset R_{!}(U\mathbin {\triangle }V) \]

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

10. 10.

    Interaction With Internal Homs of Powersets. The diagram

    
    commutes, i.e. we have an equality of sets
    \[ R_{!}([U,V]_{X})=[R_{*}(U),R_{!}(V)]_{Y}, \]

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

11. 11.

    Preservation of Colimits. We have an equality of sets

    \[ R_{!}\left(\bigcup _{i\in I}U_{i}\right)=\bigcup _{i\in I}R_{!}(U_{i}), \]

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

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

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

12. 12.

    Oplax Preservation of Limits. We have an inclusion of sets

    \[ R_{!}\left(\bigcap _{i\in I}U_{i}\right)\subset \bigcap _{i\in I}R_{!}(U_{i}), \]

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

    \[ \begin{gathered} R_{!}(U\cap V) \subset R_{!}(U)\cap R_{!}(V),\\ R_{!}(X) \subset Y, \end{gathered} \]

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

13. 13.

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

    \[ (R_{!},R^{\otimes }_{!},R^{\otimes }_{*|\mathbb {1}}) \colon (\mathcal{P}(X),\cup ,\text{Ø}) \to (\mathcal{P}(Y),\cup ,\text{Ø}), \]

    being equipped with equalities

    \[ \begin{gathered} R^{\otimes }_{*|U,V} \colon R_{!}(U)\cup R_{!}(V) \mathbin {\overset {=}{\rightarrow }}R_{!}(U\cup V),\\ R^{\otimes }_{*|\mathbb {1}} \colon \text{Ø}\mathbin {\overset {=}{\rightarrow }}\text{Ø}, \end{gathered} \]

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

14. 14.

    Symmetric Oplax Monoidality With Respect to Intersections. The direct image function of Item 1 has a symmetric oplax monoidal structure

    \[ (R_{!},R^{\otimes }_{!},R^{\otimes }_{*|\mathbb {1}}) \colon (\mathcal{P}(X),\cap ,X) \to (\mathcal{P}(Y),\cap ,Y), \]

    being equipped with inclusions

    \[ \begin{gathered} R^{\otimes }_{*|U,V} \colon R_{!}(U\cap V) \subset R_{!}(U)\cap R_{!}(V),\\ R^{\otimes }_{*|\mathbb {1}} \colon R_{!}(X) \subset Y, \end{gathered} \]

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

15. 15.

    Interaction With Coproducts. Let $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X'$ and $S\colon Y\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y'$ be relations. The diagram

    
    commutes, i.e. we have
    \[ (R\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}S)_{!}(U\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}V)=R_{!}(U)\mathchoice {\mathbin {\textstyle \coprod }}{\mathbin {\textstyle \coprod }}{\mathbin {\scriptstyle \textstyle \coprod }}{\mathbin {\scriptscriptstyle \textstyle \coprod }}S_{!}(V) \]

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

16. 16.

    Interaction With Products. Let $R\colon X\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X'$ and $S\colon Y\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}Y'$ be relations. The diagram

    
    commutes, i.e. we have
    \[ (R\boxtimes _{X\times Y}S)_{!}(U\boxtimes _{X\times Y}V)=R_{!}(U)\boxtimes _{X'\times Y'}S_{!}(V) \]

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

17. 17.

    Relation to Codirect Images. We have

    \[ R_{!}(U)=R_{*}(U^{\textsf{c}})^{\textsf{c}} \]

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