The Clowder Project

Let $X$ be a set.

1. Functoriality. The assignments $U,V,\webleft (U,V\webright )\mapsto U\cap V$ define functors
   \[ \begin{array}{ccc} U\cap -\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\cap V\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -_{1}\cap -_{2}\colon \mkern -15mu & \webleft (\mathcal{P}\webleft (X\webright )\times \mathcal{P}\webleft (X\webright ),\subset \times \subset \webright ) \mkern -17.5mu& {}\mathbin {\to }\webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \end{array} \]

   In particular, the following statements hold for each $U,V,A,B\in \mathcal{P}\webleft (X\webright )$:

   1. If $U\subset A$, then $U\cap V\subset A\cap V$.
   2. If $V\subset B$, then $U\cap V\subset U\cap B$.
   3. If $U\subset A$ and $V\subset B$, then $U\cap V\subset A\cap B$.
2. Adjointness. We have adjunctions
   
   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,\webleft [V,W\webright ]_{X}\webright ),\\ \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \textup{Hom}_{\mathcal{P}\webleft (X\webright )}\webleft (V,\webleft [U,W\webright ]_{X}\webright ), \end{align*}

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

   \[ \webleft [-_{1},-_{2}\webright ]_{X}\colon \mathcal{P}\webleft (X\webright )\mkern -0.0mu^{\mathsf{op}}\times \mathcal{P}\webleft (X\webright ) \to \mathcal{P}\webleft (X\webright ) \]

   is the bifunctor of Section 2.4.7. In particular, the following statements hold for each $U,V,W\in \mathcal{P}\webleft (X\webright )$:

   1. The following conditions are equivalent:
      1. We have $U\cap V\subset W$.
      2. We have $U\subset \webleft [V,W\webright ]_{X}$.
   2. The following conditions are equivalent:
      1. We have $U\cap V\subset W$.
      2. We have $V\subset \webleft [U,W\webright ]_{X}$.
3. Associativity. The diagram
   

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

   \[ \webleft (U\cap V\webright )\cap W=U\cap \webleft (V\cap W\webright ) \]

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

4. Unitality. The diagrams
   
   commute, i.e. we have equalities of sets
   \begin{align*} X\cap U & = U,\\ U\cap X & = U \end{align*}

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

5. Commutativity. The diagram
   

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

   \[ U\cap V= V\cap U \]

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

6. Annihilation With the Empty Set. The diagrams
   
   commute, i.e. we have equalities of sets
   \begin{align*} \text{Ø}\cap X & = \text{Ø},\\ X\cap \text{Ø}& = \text{Ø}\end{align*}

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

7. Distributivity of Unions Over Intersections. The diagrams
   
   commute, i.e. we have equalities of sets
   \begin{align*} U\cup \webleft (V\cap W\webright ) & = \webleft (U\cup V\webright )\cap \webleft (U\cup W\webright ),\\ \webleft (U\cap V\webright )\cup W & = \webleft (U\cup W\webright )\cap \webleft (V\cup W\webright ) \end{align*}

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

8. Distributivity of Intersections Over Unions. The diagrams
   
   commute, i.e. 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 $U,V,W\in \mathcal{P}\webleft (X\webright )$.

9. Idempotency. The diagram
   

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

   \[ U\cap U=U \]

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

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

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

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

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

12. Interaction With Direct Images. Let $f\colon X\to Y$ be a function. We have a natural transformation
    

    with components

    \[ f_{*}\webleft (U\cap V\webright )\subset f_{*}\webleft (U\webright )\cap f_{*}\webleft (V\webright ) \]

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

13. Interaction With Inverse Images. Let $f\colon X\to Y$ be a function. 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 $U,V\in \mathcal{P}\webleft (Y\webright )$.

14. Interaction With Direct Images With Compact Support. Let $f\colon X\to Y$ be a function. The diagram
    

    commutes, i.e. we have

    \[ f_{!}\webleft (U\webright )\cap f_{!}\webleft (V\webright )=f_{!}\webleft (U\cap V\webright ) \]

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

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