The Clowder Project

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

1. Functoriality. The assignment $V\mapsto R_{-1}\webleft (V\webright )$ defines a functor
   \[ R_{-1}\colon \webleft (\mathcal{P}\webleft (B\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (A\webright ),\subset \webright ) \]

   where

   • Action on Objects. For each $V\in \mathcal{P}\webleft (B\webright )$, we have
     \[ \webleft [R_{-1}\webright ]\webleft (V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}R_{-1}\webleft (V\webright ). \]

   • Action on Morphisms. For each $U,V\in \mathcal{P}\webleft (B\webright )$:
     • If $U\subset V$, then $R_{-1}\webleft (U\webright )\subset R_{-1}\webleft (V\webright )$.

2. Adjointness. We have an adjunction
   
   witnessed by a bijections of sets
   \[ \textup{Hom}_{\mathcal{P}\webleft (A\webright )}\webleft (R_{*}\webleft (U\webright ),V\webright )\cong \textup{Hom}_{\mathcal{P}\webleft (A\webright )}\webleft (U,R_{-1}\webleft (V\webright )\webright ), \]

   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 $R_{*}\webleft (U\webright )\subset V$.
     • We have $U\subset R_{-1}\webleft (V\webright )$.

3. Lax Preservation of Colimits. We have an inclusion of sets
   \[ \bigcup _{i\in I}R_{-1}\webleft (U_{i}\webright )\subset R_{-1}\webleft (\bigcup _{i\in I}U_{i}\webright ), \]

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

   \[ \begin{gathered} R_{-1}\webleft (U\webright )\cup R_{-1}\webleft (V\webright ) \subset R_{-1}\webleft (U\cup V\webright ),\\ \emptyset \subset R_{-1}\webleft (\emptyset \webright ), \end{gathered} \]

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

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

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

   \[ \begin{gathered} R_{-1}\webleft (U\cap V\webright ) = R_{-1}\webleft (U\webright )\cap R_{-1}\webleft (V\webright ),\\ R_{-1}\webleft (B\webright ) = B, \end{gathered} \]

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

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

   being equipped with inclusions

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

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

6. Symmetric Strict Monoidality With Respect to Intersections. The direct image function of Item 1 has a symmetric strict monoidal structure
   \[ \webleft (R_{-1},R^{\otimes }_{-1},R^{\otimes }_{-1|\mathbb {1}}\webright ) \colon \webleft (\mathcal{P}\webleft (A\webright ),\cap ,A\webright ) \to \webleft (\mathcal{P}\webleft (B\webright ),\cap ,B\webright ), \]

   being equipped with equalities

   \[ \begin{gathered} R^{\otimes }_{-1|U,V} \colon R_{-1}\webleft (U\cap V\webright ) \mathbin {\overset {=}{\rightarrow }}R_{-1}\webleft (U\webright )\cap R_{-1}\webleft (V\webright ),\\ R^{\otimes }_{-1|\mathbb {1}} \colon R_{-1}\webleft (A\webright ) \mathbin {\overset {=}{\rightarrow }}B, \end{gathered} \]

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

7. Interaction With Weak Inverse Images I. We have
   \[ R_{-1}\webleft (V\webright )=A\setminus R^{-1}\webleft (B\setminus V\webright ) \]

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

8. Interaction With Weak Inverse Images II. Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation from $A$ to $B$.
   1. If $R$ is a total relation, then we have an inclusion of sets
      \[ R_{-1}\webleft (V\webright ) \subset R^{-1}\webleft (V\webright ) \]

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

   2. If $R$ is total and functional, then the above inclusion is in fact an equality.
   3. Conversely, if we have $R_{-1}=R^{-1}$, then $R$ is total and functional.

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

1. Functionality I. The assignment $R\mapsto R_{-1}$ defines a function
   \[ \webleft (-\webright )_{-1}\colon \mathsf{Sets}\webleft (A,B\webright ) \to \mathsf{Sets}\webleft (\mathcal{P}\webleft (A\webright ),\mathcal{P}\webleft (B\webright )\webright ). \]
2. Functionality II. The assignment $R\mapsto R_{-1}$ defines a function
   \[ \webleft (-\webright )_{-1}\colon \mathsf{Sets}\webleft (A,B\webright ) \to \mathsf{Pos}\webleft (\webleft (\mathcal{P}\webleft (A\webright ),\subset \webright ),\webleft (\mathcal{P}\webleft (B\webright ),\subset \webright )\webright ). \]
3. Interaction With Identities. For each $A\in \text{Obj}\webleft (\mathsf{Sets}\webright )$, we have
   \[ \webleft (\text{id}_{A}\webright )_{-1}=\text{id}_{\mathcal{P}\webleft (A\webright )}. \]
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