Proposition 2.3.7.1.2 (01GA): Properties of Intersections of Families of Subsets

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Proposition 2.3.7.1.2 ▶ Properties of Intersections of Families of Subsets

Let $X$ be a set.

Functoriality. The assignment $\mathcal{U}\mapsto \bigcap _{U\in \mathcal{U}}U$ defines a functor
\[ \bigcap \colon \webleft (\mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright ),\supset \webright )\to \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright ). \]

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 $\displaystyle \bigcap _{V\in \mathcal{V}}V\subset \bigcap _{U\in \mathcal{U}}U$.
Oplax Associativity. We have a natural transformation

with components

\[ \bigcap _{A\in \mathcal{A}}\webleft(\bigcap _{U\in A}U\webright)\subset \bigcap _{U\in \bigcap _{A\in \mathcal{A}}A}U \]

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

\[ \bigcap _{V\in \webleft\{ U\webright\} }V=U. \]

for each $U\in \mathcal{P}\webleft (X\webright )$.
Oplax Right Unitality. The diagram

does not commute in general, i.e. we may have

\[ \bigcap _{\webleft\{ x\webright\} \in \chi _{X}\webleft (U\webright )}\webleft\{ x\webright\} \neq U \]

in general, where $U\in \mathcal{P}\webleft (X\webright )$. However, when $U$ is nonempty, we have

\[ \bigcap _{\webleft\{ x\webright\} \in \chi _{X}\webleft (U\webright )}\webleft\{ x\webright\} \subset U. \]
Interaction With Unions I. The diagram

commutes, i.e. we have

\[ \bigcap _{W\in \mathcal{U}\cup \mathcal{V}}W=\webleft(\bigcap _{U\in \mathcal{U}}U\webright)\cap \webleft(\bigcap _{V\in \mathcal{V}}V\webright) \]

for each $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
Interaction With Unions II. The diagram
commute, i.e. we have
\begin{align*} U\cup \webleft(\bigcap _{V\in \mathcal{V}}V\webright) & = \bigcap _{V\in \mathcal{V}}\webleft (U\cup V\webright ),\\ \webleft(\bigcap _{U\in \mathcal{U}}U\webright)\cup V & = \bigcap _{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

\[ \webleft(\bigcap _{U\in \mathcal{U}}U\webright)\cap \webleft(\bigcap _{V\in \mathcal{V}}V\webright)\subset \bigcap _{W\in \mathcal{U}\cap \mathcal{V}}W \]

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(\bigcap _{V\in \mathcal{V}}V\webright) & = \bigcap _{V\in \mathcal{V}}\webleft (U\cup V\webright ),\\ \webleft(\bigcap _{U\in \mathcal{U}}U\webright)\cup V & = \bigcap _{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

\[ \bigcap _{W\in \mathcal{U}\setminus \mathcal{V}}W\neq \webleft(\bigcap _{U\in \mathcal{U}}U\webright)\setminus \webleft(\bigcap _{V\in \mathcal{V}}V\webright) \]

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

\[ \bigcap _{W\in \mathcal{U}^{\textsf{c}}}W\neq \bigcap _{U\in \mathcal{U}}U^{\textsf{c}} \]

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

\[ \webleft(\bigcap _{U\in \mathcal{U}}U\webright)^{\textsf{c}}=\bigcup _{U\in \mathcal{U}}U^{\textsf{c}} \]

for each $\mathcal{U}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$.
Interaction With Complements III. 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 Symmetric Differences. The diagram

does not commute in general, i.e. we may have

\[ \bigcap _{W\in \mathcal{U}\mathbin {\triangle }\mathcal{V}}W\neq \webleft(\bigcap _{U\in \mathcal{U}}U\webright)\mathbin {\triangle }\webleft(\bigcap _{V\in \mathcal{V}}V\webright) \]

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

\[ \bigcap _{W\in \webleft [\mathcal{U},\mathcal{V}\webright ]_{\mathcal{P}\webleft (X\webright )}}W\neq \webleft[\bigcap _{U\in \mathcal{U}}U,\bigcap _{V\in \mathcal{V}}V\webright]_{X} \]

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

\[ \webleft[\bigcap _{U\in \mathcal{U}}U,V\webright]_{X}= \bigcup _{U\in \mathcal{U}}\webleft [U,V\webright ]_{X} \]

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

\[ \webleft[U,\bigcap _{V\in \mathcal{V}}V\webright]_{X}= \bigcap _{V\in \mathcal{V}}\webleft [U,V\webright ]_{X} \]

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

\[ \bigcap _{U\in \mathcal{U}}f_{*}\webleft (U\webright )=\bigcap _{V\in f_{*}\webleft (\mathcal{U}\webright )}V \]

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

\[ \bigcap _{V\in \mathcal{V}}f^{-1}\webleft (V\webright )=\bigcap _{U\in f^{-1}\webleft (\mathcal{U}\webright )}U \]

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

\[ \bigcap _{U\in \mathcal{U}}f_{!}\webleft (U\webright )=\bigcap _{V\in f_{!}\webleft (\mathcal{U}\webright )}V \]

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 Unions 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 Unions 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.

Proof of Proposition 2.3.6.1.2.

Item 1: Functoriality

Since $\mathcal{P}\webleft (X\webright )$ is posetal, it suffices to prove the condition $\webleft (\star \webright )$. So let $\mathcal{U},\mathcal{V}\in \mathcal{P}\webleft (\mathcal{P}\webleft (X\webright )\webright )$ with $\mathcal{U}\subset \mathcal{V}$. We claim that

\[ \bigcap _{V\in \mathcal{V}}V\subset \bigcap _{U\in \mathcal{U}}U. \]

Indeed, if $x\in \bigcap _{V\in \mathcal{V}}V$, then $x\in V$ for all $V\in \mathcal{V}$. But since $\mathcal{U}\subset \mathcal{V}$, it follows that $x\in U$ for all $U\in \mathcal{U}$ as well. Thus $x\in \bigcap _{U\in \mathcal{U}}U$, which gives our desired inclusion.

Item 2: Oplax Associativity

We have

\begin{align*} \bigcap _{A\in \mathcal{A}}\webleft(\bigcap _{U\in A}U\webright) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $A\in \mathcal{A}$,}\\ & \text{we have $x\in \bigcap _{U\in A}U$} \end{aligned} \webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $A\in \mathcal{A}$ and each}\\ & \text{$U\in A$, we have $x\in U$} \end{aligned} \webright\} \\ & = \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $U\in \displaystyle \bigcup _{A\in \mathcal{A}}A$,}\\ & \text{we have $x\in U$} \end{aligned} \webright\} \\ & \subset \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $U\in \displaystyle \bigcap _{A\in \mathcal{A}}A$,}\\ & \text{we have $x\in U$} \end{aligned} \webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcap _{U\in {\scriptsize \displaystyle \bigcap _{A\in \mathcal{A}}A}}U. \end{align*}

Since $\mathcal{P}\webleft (X\webright )$ is posetal, naturality is automatic (). This finishes the proof.

Item 3: Left Unitality

We have

\begin{align*} \bigcap _{V\in \webleft\{ U\webright\} }V & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $V\in \webleft\{ U\webright\} $,}\\ & \text{we have $x\in U$} \end{aligned} \webright\} \\ & = \webleft\{ x\in X\ \middle |\ x\in U \webright\} \\ & = U.\end{align*}

This finishes the proof.

Item 4: Oplax Right Unitality

If $U=\text{Ø}$, then we have

\begin{align*} \bigcap _{\webleft\{ u\webright\} \in \chi _{X}\webleft (U\webright )}\webleft\{ u\webright\} & = \bigcap _{\webleft\{ u\webright\} \in \text{Ø}}\webleft\{ u\webright\} \\ & = X,\end{align*}

so $\bigcap _{\webleft\{ u\webright\} \in \chi _{X}\webleft (U\webright )}\webleft\{ u\webright\} =X\neq \text{Ø}=U$. When $U$ is nonempty, we have two cases:

If $U$ is a singleton, say $U=\webleft\{ u\webright\} $, we have
\begin{align*} \bigcap _{\webleft\{ u\webright\} \in \chi _{X}\webleft (U\webright )}\webleft\{ u\webright\} & = \webleft\{ u\webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}U.\end{align*}
If $U$ contains at least two elements, we have
\begin{align*} \bigcap _{\webleft\{ u\webright\} \in \chi _{X}\webleft (U\webright )}\webleft\{ u\webright\} & = \text{Ø}\\ & \subset U.\end{align*}

This finishes the proof.

Item 5: Interaction With Unions I

We have

\begin{align*} \bigcap _{W\in \mathcal{U}\cup \mathcal{V}}W & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $W\in \mathcal{U}\cup \mathcal{V}$,}\\ & \text{we have $x\in W$} \end{aligned} \webright\} \\ & = \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $W\in \mathcal{U}$ and each}\\ & \text{$W\in \mathcal{V}$, we have $x\in W$} \end{aligned} \webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $W\in \mathcal{U}$,}\\ & \text{we have $x\in W$} \end{aligned} \webright\} \\ & \phantom{={}}\mkern 4mu\cap \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $W\in \mathcal{V}$,}\\ & \text{we have $x\in W$} \end{aligned} \webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft(\bigcap _{W\in \mathcal{U}}W\webright)\cap \webleft(\bigcap _{W\in \mathcal{V}}W\webright)\\ & = \webleft(\bigcap _{U\in \mathcal{U}}U\webright)\cap \webleft(\bigcap _{V\in \mathcal{V}}V\webright). \end{align*}

This finishes the proof.

Item 6: Interaction With Unions II

Omitted.

Item 7: Interaction With Intersections I

We have

\begin{align*} \webleft(\bigcap _{U\in \mathcal{U}}U\webright)\cap \webleft(\bigcap _{V\in \mathcal{V}}V\webright) & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $U\in \mathcal{U}$,}\\ & \text{we have $x\in U$} \end{aligned} \webright\} \\ & \phantom{={}}\mkern 4mu\cup \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $V\in \mathcal{V}$,}\\ & \text{we have $x\in V$} \end{aligned} \webright\} \\ & = \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $W\in \mathcal{U}\cap \mathcal{V}$,}\\ & \text{we have $x\in W$} \end{aligned} \webright\} \\ & \subset \webleft\{ x\in X\ \middle |\ \begin{aligned} & \text{for each $W\in \mathcal{U}\cup \mathcal{V}$,}\\ & \text{we have $x\in W$} \end{aligned} \webright\} \\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcap _{W\in \mathcal{U}\cap \mathcal{V}}W.\end{align*}

Since $\mathcal{P}\webleft (X\webright )$ is posetal, naturality is automatic (). This finishes the proof.

Item 8: Interaction With Intersections II

Omitted.

Item 9: Interaction With Differences

Let $X=\webleft\{ 0,1\webright\} $, let $\mathcal{U}=\webleft\{ \webleft\{ 0\webright\} ,\webleft\{ 0,1\webright\} \webright\} $, and let $\mathcal{V}=\webleft\{ \webleft\{ 0\webright\} \webright\} $. We have

\begin{align*} \bigcap _{W\in \mathcal{U}\setminus \mathcal{V}}U & = \bigcap _{W\in \webleft\{ \webleft\{ 0,1\webright\} \webright\} }W\\ & = \webleft\{ 0,1\webright\} , \end{align*}

whereas

\begin{align*} \webleft(\bigcap _{U\in \mathcal{U}}U\webright)\setminus \webleft(\bigcap _{V\in \mathcal{V}}V\webright) & = \webleft\{ 0\webright\} \setminus \webleft\{ 0\webright\} \\ & = \text{Ø}. \end{align*}

Thus we have

\[ \bigcap _{W\in \mathcal{U}\setminus \mathcal{V}}W=\webleft\{ 0,1\webright\} \neq \text{Ø}=\webleft(\bigcap _{U\in \mathcal{U}}U\webright)\setminus \webleft(\bigcap _{V\in \mathcal{V}}V\webright). \]

This finishes the proof.

Item 10: Interaction With Complements I

Let $X=\webleft\{ 0,1\webright\} $ and let $\mathcal{U}=\webleft\{ \webleft\{ 0\webright\} \webright\} $. We have

\begin{align*} \bigcap _{W\in \mathcal{U}^{\textsf{c}}}U & = \bigcap _{W\in \webleft\{ \text{Ø},\webleft\{ 1\webright\} ,\webleft\{ 0,1\webright\} \webright\} }W\\ & = \text{Ø}, \end{align*}

whereas

\begin{align*} \bigcap _{U\in \mathcal{U}}U^{\textsf{c}} & = \webleft\{ 0\webright\} ^{\textsf{c}}\\ & = \webleft\{ 1\webright\} . \end{align*}

Thus we have

\[ \bigcap _{W\in \mathcal{U}^{\textsf{c}}}U=\text{Ø}\neq \webleft\{ 1\webright\} =\bigcap _{U\in \mathcal{U}}U^{\textsf{c}}. \]

This finishes the proof.

Item 11: Interaction With Complements II

This is a repetition of Item 12 of Proposition 2.3.6.1.2 and is proved there.

Item 12: Interaction With Complements III

This is a repetition of Item 11 of Proposition 2.3.6.1.2 and is proved there.

Item 13: Interaction With Symmetric Differences

Let $X=\webleft\{ 0,1\webright\} $, let $\mathcal{U}=\webleft\{ \webleft\{ 0,1\webright\} \webright\} $, and let $\mathcal{V}=\webleft\{ \webleft\{ 0\webright\} ,\webleft\{ 0,1\webright\} \webright\} $. We have

\begin{align*} \bigcap _{W\in \mathcal{U}\mathbin {\triangle }\mathcal{V}}W & = \bigcap _{W\in \webleft\{ \webleft\{ 0\webright\} \webright\} }W\\ & = \webleft\{ 0\webright\} , \end{align*}

whereas

\begin{align*} \webleft(\bigcap _{U\in \mathcal{U}}U\webright)\mathbin {\triangle }\webleft(\bigcap _{V\in \mathcal{V}}V\webright) & = \webleft\{ 0,1\webright\} \mathbin {\triangle }\webleft\{ 0\webright\} \\ & = \text{Ø}, \end{align*}

Thus we have

\[ \bigcap _{W\in \mathcal{U}\mathbin {\triangle }\mathcal{V}}W=\webleft\{ 0\webright\} \neq \text{Ø}=\webleft(\bigcap _{U\in \mathcal{U}}U\webright)\mathbin {\triangle }\webleft(\bigcap _{V\in \mathcal{V}}V\webright). \]

This finishes the proof.

Item 14: Interaction With Internal Homs I

This is a repetition of Item 10 of Proposition 2.4.7.1.3 and is proved there.

Item 15: Interaction With Internal Homs II

This is a repetition of Item 11 of Proposition 2.4.7.1.3 and is proved there.

Item 16: Interaction With Internal Homs III

This is a repetition of Item 12 of Proposition 2.4.7.1.3 and is proved there.

Item 17: Interaction With Direct Images

This is a repetition of Item 4 of Proposition 2.6.1.1.4 and is proved there.

Item 18: Interaction With Inverse Images

This is a repetition of Item 4 of Proposition 2.6.2.1.3 and is proved there.

Item 19: Interaction With Direct Images With Compact Support