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{gather*} \begin{aligned} U\cup - & \colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ -\cup V & \colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ) \to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ),\\ \end{aligned}\\ -_{1}\cup -_{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 $-_{1}\cup -_{2}$ 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
     \[ \webleft [-_{1}\cup -_{2}\webright ]\webleft (U,V\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U\cup V. \]

   • 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

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

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

     \[ U\cup V\subset U'\cup V' \]

     i.e. where we have

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

2. Via Intersections and Symmetric Differences. We have an equality of sets
   \[ U\cup V=\webleft (U\mathbin {\triangle }V\webright )\mathbin {\triangle }\webleft (U\cap V\webright ) \]

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

3. Associativity. We have an equality of sets
   \[ \webleft (U\cup V\webright )\cup W = U\cup \webleft (V\cup W\webright ) \]

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

4. Unitality. We have equalities of sets
   \begin{align*} U\cup \emptyset & = U,\\ \emptyset \cup U & = 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
   \[ U\cup V = V\cup U \]

   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
   \[ U\cup U=U \]

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

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

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

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

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

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

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