The Clowder Project

Let $X$ be a set.

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

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

   1. If $U\subset A$, then $U\cup V\subset A\cup V$.
   2. If $V\subset B$, then $U\cup V\subset U\cup B$.
   3. If $U\subset A$ and $V\subset B$, then $U\cup V\subset A\cup B$.
2. Associativity. The diagram
   

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

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

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

3. Unitality. The diagrams
   
   commute, i.e. we have equalities of sets
   \begin{align*} \text{Ø}\cup U & = U,\\ U\cup \text{Ø}& = U \end{align*}

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

4. Commutativity. The diagram
   

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

   \[ U\cup V = V\cup U \]

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

5. Annihilation With $X$. The diagrams
   
   commute, i.e. we have equalities of sets
   \begin{align*} U\cup X & = X,\\ X\cup V & = X \end{align*}

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

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

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

8. Idempotency. The diagram
   

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

   \[ U\cup U=U \]

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

9. Via Intersections and Symmetric Differences. The diagram
   

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

   \[ U\cup V=\webleft (U\mathbin {\triangle }V\webright )\mathbin {\triangle }\webleft (U\cap V\webright ) \]

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

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

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

11. Interaction With Characteristic Functions II. We have
    \[ \chi _{U\cup V}=\chi _{U}+\chi _{V}-\chi _{U\cap V} \]

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

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

    commutes, i.e. we have

    \[ f_{*}\webleft (U\cup V\webright )=f_{*}\webleft (U\webright )\cup f_{*}\webleft (V\webright ) \]

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

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

    commutes, i.e. we have

    \[ f^{-1}\webleft (U\cup V\webright )=f^{-1}\webleft (U\webright )\cup f^{-1}\webleft (V\webright ) \]

    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. We have a natural transformation
    

    with components

    \[ f_{!}\webleft (U\webright )\cup f_{!}\webleft (V\webright )\subset f_{!}\webleft (U\cup V\webright ) \]

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

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