The Clowder Project

Let $X$ be a set.

1. Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto U\cap V$ define functors
   \begin{gather*} \begin{aligned} U\cap - & \colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\cap V & \colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ \end{aligned}\\ -_{1}\cap -_{2} \colon \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \subset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ), \end{gather*}

   where $-_{1}\cap -_{2}$ is the functor where

   • Action on Objects. For each $\webleft (U,V\webright )\in \mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright )$, we have
     \[ \webleft [-_{1}\cap -_{2}\webright ]\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U\cap V. \]

   • Action on Morphisms. For each pair of morphisms
     \begin{align*} \iota _{U} & \colon U\hookrightarrow U',\\ \iota _{V} & \colon V\hookrightarrow V’ \end{align*}

     of $\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright )$, the image

     \[ \iota _{U}\cap \iota _{V}\colon U\cap V\hookrightarrow U'\cap V' \]

     of $\webleft (\iota _{U},\iota _{V}\webright )$ by $\cap $ is the inclusion

     \[ U\cap V\subset U'\cap V' \]

     i.e. where we have

     • If $U\subset U'$ and $V\subset V'$, then $U\cap V\subset U'\cap V'$.
   and where $U\cap -$ and $-\cap V$ are the partial functors of $-_{1}\cap -_{2}$ at $U,V\in \mathcal{P}\webleft (X\webright )$.

2. Adjointness. We have adjunctions
   
   where
   \[ \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (-_{1},-_{2}\webright )\colon \mathcal{P}\webleft (X\webright )\mkern -0.0mu^{\mathsf{op}}\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \]

   is the bifunctor defined by^[1]

   \[ \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (X\setminus U\webright )\cup V \]

   witnessed by bijections

   \begin{align*} \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,\mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (V,W\webright )\webright ),\\ \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (V,\mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,W\webright )\webright ), \end{align*}

   natural in $U,V,W\in \mathcal{P}\webleft (X\webright )$, i.e. where:

   1. The following conditions are equivalent:
      1. We have $U\cap V\subset W$.
      2. We have $U\subset \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (V,W\webright )$.
      3. We have $U\subset \webleft (X\setminus V\webright )\cup W$.
   2. The following conditions are equivalent:
      1. We have $V\cap U\subset W$.
      2. We have $V\subset \mathbf{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U,W\webright )$.
      3. We have $V\subset \webleft (X\setminus U\webright )\cup W$.
3. Associativity. We have an equality of sets
   \[ \webleft (U\cap V\webright )\cap W=U\cap \webleft (V\cap W\webright ) \]

   for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

4. Unitality. Let $X$ be a set and let $U\in \mathcal{P}\webleft (X\webright )$. We have equalities of sets
   \begin{align*} X\cap U & = U,\\ U\cap X & = U \end{align*}

   for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U\in \mathcal{P}\webleft (X\webright )$.

5. Commutativity. We have an equality of sets
   \[ U\cap V= V\cap U \]

   for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.

6. Idempotency. We have an equality of sets
   \[ U\cap U=U \]

   for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U\in \mathcal{P}\webleft (X\webright )$.

7. Distributivity Over Unions. We have equalities of sets
   \begin{align*} U\cap \webleft (V\cup W\webright ) & = \webleft (U\cap V\webright )\cup \webleft (U\cap W\webright ),\\ \webleft (U\cup V\webright )\cap W & = \webleft (U\cap W\webright )\cup \webleft (V\cap W\webright ) \end{align*}

   for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V,W\in \mathcal{P}\webleft (X\webright )$.

8. Annihilation With the Empty Set. We have an equality of sets
   \begin{align*} \emptyset \cap X & = \emptyset ,\\ X\cap \emptyset & = \emptyset \end{align*}

   for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U\in \mathcal{P}\webleft (X\webright )$.

9. Interaction With Characteristic Functions I. We have
   \[ \chi _{U\cap V}=\chi _{U}\chi _{V} \]

   for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.

10. Interaction With Characteristic Functions II. We have
    \[ \chi _{U\cap V}=\operatorname*{\text{min}}\webleft (\chi _{U},\chi _{V}\webright ) \]

    for each $X\in \text{Obj}\webleft (\mathsf{Sets}\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.

11. Interaction With Powersets and Monoids With Zero. The quadruple $\webleft (\webleft (\mathcal{P}\webleft (X\webright ),\emptyset \webright ),\cap ,X\webright )$ is a commutative monoid with zero.
12. Interaction With Powersets and Semirings. The quintuple $\webleft (\mathcal{P}\webleft (X\webright ),\cup ,\cap ,\emptyset ,X\webright )$ is an idempotent commutative semiring.