The Clowder Project

Let $X$ be a set.

1. 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. (a)

      If $U\subset A$, then $U\cap V\subset A\cap V$.

   2. (b)

      If $V\subset B$, then $U\cap V\subset U\cap B$.

   3. (c)

      If $U\subset A$ and $V\subset B$, then $U\cap V\subset A\cap B$.

2. 2.

   Adjointness. We have adjunctions

   
   witnessed by bijections
   \begin{align*} \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U,\webleft [V,W\webright ]_{X}\webright ),\\ \operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (U\cap V,W\webright ) & \cong \operatorname {\mathrm{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 4.4.7. In particular, the following statements hold for each $U,V,W\in \mathcal{P}\webleft (X\webright )$:

   1. (a)

      The following conditions are equivalent:

      1. (i)

         We have $U\cap V\subset W$.

      2. (ii)

         We have $U\subset \webleft [V,W\webright ]_{X}$.

   2. (b)

      The following conditions are equivalent:

      1. (i)

         We have $U\cap V\subset W$.

      2. (ii)

         We have $V\subset \webleft [U,W\webright ]_{X}$.

3. 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. 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. 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. 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. 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. 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. 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. 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. 11.

    Interaction With Characteristic Functions II. We have

    \[ \chi _{U\cap V}=\operatorname*{\operatorname {\mathrm{min}}}\webleft (\chi _{U},\chi _{V}\webright ) \]

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

12. 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. 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. 14.

    Interaction With Codirect Images. 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. 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. 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.