Item 2a(i) (01HA) — The Clowder Project

4.3.9 Binary Intersections

Let $X$ be a set and let $U,V\in \mathcal{P}(X)$.

Definition 4.3.9.1.1 ▶ Binary Intersections

The intersection of $U$ and $V$ is the set $U\cap V$ defined by

\begin{align*} U\cap V & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\bigcap _{z\in \left\{ U,V\right\} }z\\ & \mathrel {\smash {\overset {\mathclap {\scriptscriptstyle \text{def}}}=}}\left\{ x\in X\ \middle |\ \text{$x\in U$ or $x\in V$}\right\} . \end{align*}

Proposition 4.3.9.1.2 ▶ Properties of Binary Intersections

Let $X$ be a set.

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.
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:

(i)
We have $U\cap V\subset W$.

(ii)

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

(b)

The following conditions are equivalent:

(i)
We have $U\cap V\subset W$.
(ii)
We have $V\subset [U,W]_{X}$.

Associativity. The diagram

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

\[ (U\cap V)\cap W=U\cap (V\cap W) \]

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

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

Commutativity. The diagram

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

\[ U\cap V= V\cap U \]

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

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

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

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

Idempotency. The diagram

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

\[ U\cap U=U \]

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

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.

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.

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.

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.

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.

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

16.

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

Proof of Proposition 4.3.9.1.2.

Item 1: Functoriality

See [Proof Wiki Contributors, Set Intersection Preserves Subsets — Proof Wiki].

Item 2: Adjointness

See [Lin, Show that the powerset partial order is a cartesian closed category.].

Item 3: Associativity

See [Proof Wiki Contributors, Intersection Is Associative — Proof Wiki].

Item 4: Unitality

This follows from [Proof Wiki Contributors, Intersection With Subset Is Subset — Proof Wiki] and Item 5.

Item 5: Commutativity

See [Proof Wiki Contributors, Intersection Is Commutative — Proof Wiki].

Item 6: Annihilation With the Empty Set

This follows from [Proof Wiki Contributors, Intersection With Empty Set — Proof Wiki] and Item 5.

Item 7: Distributivity of Unions Over Intersections

See [Proof Wiki Contributors, Union Distributes Over Intersection — Proof Wiki].

Item 8: Distributivity of Intersections Over Unions

See [Proof Wiki Contributors, Set Intersection Distributes Over Union — Proof Wiki].

Item 9: Idempotency

See [Proof Wiki Contributors, Set Intersection Is Idempotent — Proof Wiki].

Item 10: Interaction With Characteristic Functions I

See [Proof Wiki Contributors, Characteristic Function of Intersection — Proof Wiki].

Item 11: Interaction With Characteristic Functions II

See [Proof Wiki Contributors, Characteristic Function of Intersection — Proof Wiki].

Item 12: Interaction With Direct Images

See [Proof Wiki Contributors, Image of Intersection Under Mapping — Proof Wiki].

Item 13: Interaction With Inverse Images

See [Proof Wiki Contributors, Preimage of Intersection Under Mapping — Proof Wiki].

Item 14: Interaction With Codirect Images

This is a repetition of Item 6 of Proposition 4.6.3.1.7 and is proved there.

Item 15: Interaction With Powersets and Monoids With Zero

This follows from Item 3, Item 4, Item 5, and Item 6.

Item 16: Interaction With Powersets and Semirings

This follows from Item 2, Item 3, Item 4, and Item 8 and Item 3, Item 4, Item 5, Item 8, and Item 6 of Proposition 4.3.9.1.2.

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