Let $X$ be a set.
-
Functoriality. The assignment $\mathcal{U}\mapsto \bigcup _{U\in \mathcal{U}}U$ defines a functor
\[ \bigcup \colon \webleft (\mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \]
In particular, for each $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$, the following condition is satisfied:
- If $\mathcal{U}\subset \mathcal{V}$, then $\displaystyle \bigcup _{U\in \mathcal{U}}U\subset \bigcup _{V\in \mathcal{V}}V$.
-
Associativity. The diagram
commutes, i.e. we have
\[ \bigcup _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U=\bigcup _{A\in \mathcal{A}}\webleft (\bigcup _{U\in A}U\webright ) \]for each $\mathcal{A}\in \mathcal{P}\webleft (\mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )\webright )$.
-
Left Unitality. The diagram
commutes, i.e. we have
\[ \bigcup _{V\in \webleft\{ U\webright\} }V=U \]for each $U\in \mathcal{P}\webleft (X\webright )$.
-
Right Unitality. The diagram
commutes, i.e. we have
\[ \bigcup _{\webleft\{ u\webright\} \in \chi _{X}\webleft (U\webright )}\webleft\{ u\webright\} =U \]for each $U\in \mathcal{P}\webleft (X\webright )$.
-
Interaction With Unions I. The diagram
commutes, i.e. we have
\[ \bigcup _{W\in \mathcal{U}\cup \mathcal{V}}W=\webleft(\bigcup _{U\in \mathcal{U}}U\webright)\cup \webleft(\bigcup _{V\in \mathcal{V}}V\webright) \]for each $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
-
Interaction With Unions II. The diagrams commute, i.e. we have
\begin{align*} U\cup \webleft(\bigcup _{V\in \mathcal{V}}V\webright) & = \bigcup _{V\in \mathcal{V}}\webleft (U\cup V\webright ),\\ \webleft(\bigcup _{U\in \mathcal{U}}U\webright)\cup V & = \bigcup _{U\in \mathcal{U}}\webleft (U\cup V\webright )\end{align*}for each $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.
-
Interaction With Intersections I. We have a natural transformation
with components
\[ \bigcup _{W\in \mathcal{U}\cap \mathcal{V}}W\subset \webleft(\bigcup _{U\in \mathcal{U}}U\webright)\cap \webleft(\bigcup _{V\in \mathcal{V}}V\webright) \]for each $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
-
Interaction With Intersections II. The diagrams commute, i.e. we have
\begin{align*} U\cup \webleft(\bigcup _{V\in \mathcal{V}}V\webright) & = \bigcup _{V\in \mathcal{V}}\webleft (U\cup V\webright ),\\ \webleft(\bigcup _{U\in \mathcal{U}}U\webright)\cup V & = \bigcup _{U\in \mathcal{U}}\webleft (U\cup V\webright )\end{align*}for each $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ and each $U,V\in \mathcal{P}\webleft (X\webright )$.
-
Interaction With Differences. The diagram
does not commute in general, i.e. we may have
\[ \bigcup _{W\in \mathcal{U}\setminus \mathcal{V}}W\neq \webleft(\bigcup _{U\in \mathcal{U}}U\webright)\setminus \webleft(\bigcup _{V\in \mathcal{V}}V\webright) \]in general, where $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
-
Interaction With Complements I. The diagram
does not commute in general, i.e. we may have
\[ \bigcup _{U\in \mathcal{U}^{\textsf{c}}}U\neq \bigcup _{U\in \mathcal{U}}U^{\textsf{c}} \]in general, where $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
-
Interaction With Complements II. The diagram
commutes, i.e. we have
\[ \webleft(\bigcup _{U\in \mathcal{U}}U\webright)^{\textsf{c}}=\bigcap _{U\in \mathcal{U}}U^{\textsf{c}} \]for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
-
Interaction With Complements III. The diagram
commutes, i.e. we have
\[ \webleft(\bigcap _{U\in \mathcal{U}}U\webright)^{\textsf{c}}=\bigcup _{U\in \mathcal{U}}U^{\textsf{c}} \]for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
-
Interaction With Symmetric Differences. The diagram
does not commute in general, i.e. we may have
\[ \bigcup _{W\in \mathcal{U}\mathbin {\triangle }\mathcal{V}}W\neq \webleft(\bigcup _{U\in \mathcal{U}}U\webright)\mathbin {\triangle }\webleft(\bigcup _{V\in \mathcal{V}}V\webright) \]in general, where $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
-
Interaction With Internal Homs I. The diagram
does not commute in general, i.e. we may have
\[ \bigcup _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W\neq \webleft[\bigcup _{U\in \mathcal{U}}U,\bigcup _{V\in \mathcal{V}}V\webright]_{X} \]in general, where $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
-
Interaction With Internal Homs II. The diagram
commutes, i.e. we have
\[ \webleft[\bigcup _{U\in \mathcal{U}}U,V\webright]_{X}= \bigcap _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X} \]for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ and each $V\in \mathcal{P}\webleft (X\webright )$.
-
Interaction With Internal Homs III. The diagram
commutes, i.e. we have
\[ \webleft[U,\bigcup _{V\in \mathcal{V}}V\webright]_{X}= \bigcup _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} \]for each $U\in \mathcal{P}\webleft (X\webright )$ and each $\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
-
Interaction With Direct Images. Let $f\colon X\to Y$ be a map of sets. The diagram
commutes, i.e. we have
\[ \bigcup _{U\in \mathcal{U}}f_{*}\webleft (U\webright )=\bigcup _{V\in f_{*}\webleft (\mathcal{U}\webright )}V \]for each $\mathcal{U}\in \mathcal{P}\webleft (X\webright )$, where $f_{*}\webleft (\mathcal{U}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{*}\webright )_{*}\webleft (\mathcal{U}\webright )$.
-
Interaction With Inverse Images. Let $f\colon X\to Y$ be a map of sets. The diagram
commutes, i.e. we have
\[ \bigcup _{V\in \mathcal{V}}f^{-1}\webleft (V\webright )=\bigcup _{U\in f^{-1}\webleft (\mathcal{U}\webright )}U \]for each $\mathcal{V}\in \mathcal{P}\webleft (Y\webright )$, where $f^{-1}\webleft (\mathcal{V}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f^{-1}\webright )^{-1}\webleft (\mathcal{V}\webright )$.
-
Interaction With Direct Images With Compact Support. Let $f\colon X\to Y$ be a map of sets. The diagram
commutes, i.e. we have
\[ \bigcup _{U\in \mathcal{U}}f_{!}\webleft (U\webright )=\bigcup _{V\in f_{!}\webleft (\mathcal{U}\webright )}V \]for each $\mathcal{U}\in \mathcal{P}\webleft (X\webright )$, where $f_{!}\webleft (\mathcal{U}\webright )\mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft (f_{!}\webright )_{!}\webleft (\mathcal{U}\webright )$.
-
Interaction With Intersections of Families I. The diagram
commutes, i.e. we have
\[ \bigcap _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U=\bigcap _{A\in \mathcal{A}}\webleft(\bigcap _{U\in A}U\webright) \]for each $\mathcal{A}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
-
Interaction With Intersections of Families II. Let $X$ be a set and consider the compositions
given by
\[ \begin{aligned} \mathcal{A} & \mapsto \bigcup _{U\in {\scriptsize \displaystyle \bigcap _{A\in \mathcal{A}}}A}U,\\ \mathcal{A} & \mapsto \bigcup _{A\in \mathcal{A}}\webleft(\bigcap _{U\in A}U\webright), \end{aligned} \quad \begin{aligned} \mathcal{A} & \mapsto \bigcap _{U\in {\scriptsize \displaystyle \bigcup _{A\in \mathcal{A}}A}}U,\\ \mathcal{A} & \mapsto \bigcap _{A\in \mathcal{A}}\webleft(\bigcup _{U\in A}U\webright) \end{aligned} \]for each $\mathcal{A}\in \mathcal{P}\webleft (\mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )\webright )$. We have the following inclusions:
All other possible inclusions fail to hold in general.
