The Clowder Project

Let $X$ be a set.

1. 1.

   Functoriality. The assignments $U,V,(U,V)\mapsto U\cap V$ define functors

   \[ \begin{array}{ccc} U\cap -\colon \mkern -15mu & (\mathcal{P}(X),\subset ) \mkern -17.5mu& {}\mathbin {\to }(\mathcal{P}(X),\subset ),\\ -\cap V\colon \mkern -15mu & (\mathcal{P}(X),\subset ) \mkern -17.5mu& {}\mathbin {\to }(\mathcal{P}(X),\subset ),\\ -_{1}\cap -_{2}\colon \mkern -15mu & (\mathcal{P}(X)\times \mathcal{P}(X),\subset \times \subset ) \mkern -17.5mu& {}\mathbin {\to }(\mathcal{P}(X),\subset ). \end{array} \]

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

   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}(X)}(U\cap V,W) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(U,[V,W]_{X}),\\ \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(U\cap V,W) & \cong \operatorname {\mathrm{Hom}}_{\mathcal{P}(X)}(V,[U,W]_{X}), \end{align*}

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

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

   is the bifunctor of Section 4.4.7. In particular, the following statements hold for each $U,V,W\in \mathcal{P}(X)$:

   1. (a)

      The following conditions are equivalent:

      1. (i)

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

      2. (ii)

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

   2. (b)

      The following conditions are equivalent:

      1. (i)

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

      2. (ii)

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

3. 3.

   Associativity. The diagram

   
   commutes, i.e. we have an equality of sets
   \[ (U\cap V)\cap W=U\cap (V\cap W) \]

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

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

5. 5.

   Commutativity. The diagram

   
   commutes, i.e. we have an equality of sets
   \[ U\cap V= V\cap U \]

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

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

7. 7.

   Distributivity of Unions Over Intersections. The diagrams

   
   commute, i.e. we have equalities of sets
   \begin{align*} U\cup (V\cap W) & = (U\cup V)\cap (U\cup W),\\ (U\cap V)\cup W & = (U\cup W)\cap (V\cup W) \end{align*}

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

8. 8.

   Distributivity of Intersections Over Unions. The diagrams

   
   commute, i.e. we have equalities of sets
   \begin{align*} U\cap (V\cup W) & = (U\cap V)\cup (U\cap W),\\ (U\cup V)\cap W & = (U\cap W)\cup (V\cap W) \end{align*}

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

9. 9.

   Idempotency. The diagram

   
   commutes, i.e. we have an equality of sets
   \[ U\cap U=U \]

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

10. 10.

    Interaction With Characteristic Functions I. We have

    \[ \chi _{U\cap V}=\chi _{U}\chi _{V} \]

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

11. 11.

    Interaction With Characteristic Functions II. We have

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

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

12. 12.

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

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

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

13. 13.

    Interaction With Inverse Images. Let $f\colon X\to Y$ be a function. The diagram

    
    commutes, i.e. we have
    \[ f^{-1}(U\cap V)=f^{-1}(U)\cap f^{-1}(V) \]

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

14. 14.

    Interaction With Codirect Images. Let $f\colon X\to Y$ be a function. The diagram

    
    commutes, i.e. we have
    \[ f_{*}(U)\cap f_{*}(V)=f_{*}(U\cap V) \]

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

15. 15.

    Interaction With Powersets and Monoids With Zero. The quadruple $((\mathcal{P}(X),\text{Ø}),\cap ,X)$ is a commutative monoid with zero.

16. 16.

    Interaction With Powersets and Semirings. The quintuple $(\mathcal{P}(X),\cup ,\cap ,\text{Ø},X)$ is an idempotent commutative semiring.