Proposition 2.4.7.1.3 (01K0): Properties of Internal Homs of Powersets

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Proposition 2.4.7.1.3 ▶ Properties of Internal Homs of Powersets

Let $X$ be a set.

1. Functoriality. The assignments $U, V, (U, V) \mapsto {Hom}_{P (X)}$ define functors
$\begin{array}{ccc} [U, -]_{X} : & (P (X), \supset) & \to (P (X), \subset), \\ [-, V]_{X} : & (P (X), \subset) & \to (P (X), \subset), \\ [-_{1}, -_{2}]_{X} : & (P (X) \times P (X), \subset \times \supset) & \to (P (X), \subset) . \end{array}$

In particular, the following statements hold for each $U, V, A, B \in P (X)$ :
1. (a) If $U \subset A$ , then $[A, V]_{X} \subset [U, V]_{X}$ .
2. (b) If $V \subset B$ , then $[U, V]_{X} \subset [U, B]_{X}$ .
3. (c) If $U \subset A$ and $V \subset B$ , then $[A, V]_{X} \subset [U, B]_{X}$ .
2. Adjointness. We have adjunctions
witnessed by bijections
$\begin{aligned} {Hom}_{P (X)} (U \cap V, W) & ≅ {Hom}_{P (X)} (U, [V, W]_{X}), \\ {Hom}_{P (X)} (U \cap V, W) & ≅ {Hom}_{P (X)} (V, [U, W]_{X}) . \end{aligned}$

In particular, the following statements hold for each $U, V, W \in P (X)$ :
1. (a) The following conditions are equivalent:
  1. (i) We have $U \cap V \subset W$ .
  2. (ii) We have $U \subset [V, W]_{X}$ .
2. (b) The following conditions are equivalent:
  1. (i) We have $U \cap V \subset W$ .
  2. (ii) We have $V \subset [U, W]_{X}$ .
3. Interaction With the Empty Set I. We have
$\begin{aligned} [U, Ø]_{X} & = U^{c}, \\ [Ø, V]_{X} & = X, \end{aligned}$

natural in $U, V \in P (X)$ .
4. Interaction With $X$ . We have
$\begin{aligned} [U, X]_{X} & = X, \\ [X, V]_{X} & = V, \end{aligned}$

natural in $U, V \in P (X)$ .
5. Interaction With the Empty Set II. The functor
$D_{X} : P (X)^{op} \to P (X)$

defined by

$\begin{aligned} D_{X} & \overset{def}{=} [-, Ø]_{X} \\ = (-)^{c} \end{aligned}$

is an involutory isomorphism of categories, making $Ø$ into a dualising object for $(P (X), \cap, X, [-, -]_{X})$ in the sense of . In particular:
1. (a) The diagram
  
  commutes, i.e. we have
  
  $\underset{\overset{def}{=} [[U, Ø]_{X}, Ø]_{X}}{\underset{⏟}{D_{X} (D_{X} (U))}} = U$
  
  for each $U \in P (X)$ .
2. (b) The diagram
  
  commutes, i.e. we have
  
  $\underset{\overset{def}{=} [U \cap [V, Ø]_{X}, Ø]_{X}}{\underset{⏟}{D_{X} (U \cap D_{X} (V))}} = [U, V]_{X}$
  
  for each $U, V \in P (X)$ .
6. Interaction With the Empty Set III. Let $f : X \to Y$ be a function.
1. (a) Interaction With Direct Images. The diagram
  
  commutes, i.e. we have
  
  $f_{*} (D_{X} (U)) = D_{Y} (f_{!} (U))$
  
  for each $U \in P (X)$ .
2. (b) Interaction With Inverse Images. The diagram
  
  commutes, i.e. we have
  
  $f^{- 1} (D_{Y} (U)) = D_{X} (f^{- 1} (U))$
  
  for each $U \in P (X)$ .
3. (c) Interaction With Direct Images With Compact Support. The diagram
  
  commutes, i.e. we have
  
  $f_{!} (D_{X} (U)) = D_{Y} (f_{*} (U))$
  
  for each $U \in P (X)$ .
7. Interaction With Unions of Families of Subsets I. The diagram

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

$⋃_{W \in [U, V]_{P (X)}} W \neq [⋃_{U \in U} U, ⋃_{V \in V} V]_{X}$

in general, where $U \in P (P (X))$ .
8. Interaction With Unions of Families of Subsets II. The diagram

commutes, i.e. we have

$[⋃_{U \in U} U, V]_{X} = ⋂_{U \in U} [U, V]_{X}$

for each $U \in P (P (X))$ and each $V \in P (X)$ .
9. Interaction With Unions of Families of Subsets III. The diagram

commutes, i.e. we have

$[U, ⋃_{V \in V} V]_{X} = ⋃_{V \in V} [U, V]_{X}$

for each $U \in P (X)$ and each $V \in P (P (X))$ .
10. Interaction With Intersections of Families of Subsets I. The diagram

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

$⋂_{W \in [U, V]_{P (X)}} W \neq [⋂_{U \in U} U, ⋂_{V \in V} V]_{X}$

in general, where $U \in P (P (X))$ .
11. Interaction With Intersections of Families of Subsets II. The diagram

commutes, i.e. we have

$[⋂_{U \in U} U, V]_{X} = ⋃_{U \in U} [U, V]_{X}$

for each $U \in P (P (X))$ and each $V \in P (X)$ .
12. Interaction With Intersections of Families of Subsets III. The diagram

commutes, i.e. we have

$[U, ⋂_{V \in V} V]_{X} = ⋂_{V \in V} [U, V]_{X}$

for each $U \in P (X)$ and each $V \in P (P (X))$ .
13. Interaction With Binary Unions. We have equalities of sets
$\begin{aligned} [U \cap V, W]_{X} & = [U, W]_{X} \cup [V, W]_{X}, \\ [U, V \cap W]_{X} & = [U, V]_{X} \cap [U, W]_{X} \end{aligned}$

for each $U, V, W \in P (X)$ .
14. Interaction With Binary Intersections. We have equalities of sets
$\begin{aligned} [U \cup V, W]_{X} & = [U, W]_{X} \cap [V, W]_{X}, \\ [U, V \cup W]_{X} & = [U, V]_{X} \cup [U, W]_{X} \end{aligned}$

for each $U, V, W \in P (X)$ .
15. Interaction With Differences. We have equalities of sets
$\begin{aligned} [U ∖ V, W]_{X} & = [U, W]_{X} \cup [V^{c}, W]_{X} \\ = [U, W]_{X} \cup [U, V]_{X}, \\ [U, V ∖ W]_{X} & = [U, V]_{X} ∖ (U \cap W) \end{aligned}$

for each $U, V, W \in P (X)$ .
16. Interaction With Complements. We have equalities of sets
$\begin{aligned} [U^{c}, V]_{X} & = U \cup V, \\ [U, V^{c}]_{X} & = U \cap V, \\ [U, V]_{X}^{c} & = U ∖ V \end{aligned}$

for each $U, V \in P (X)$ .
17. Interaction With Characteristic Functions. We have
$χ_{[U, V]_{P (X)}} (x) = max (1 - χ_{U} (\mod 2), χ_{V})$

for each $U, V \in P (X)$ .
18. Interaction With Direct Images. Let $f : X \to Y$ be a function. The diagram

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

$f_{*} ([U, V]_{X}) = [f_{!} (U), f_{*} (V)]_{Y},$

natural in $U, V \in P (X)$ .
19. Interaction With Inverse Images. Let $f : X \to Y$ be a function. The diagram

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

$f^{- 1} ([U, V]_{Y}) = [f^{- 1} (U), f^{- 1} (V)]_{X},$

natural in $U, V \in P (X)$ .
20. Interaction With Direct Images With Compact Support. Let $f : X \to Y$ be a function. We have a natural transformation

with components

$[f_{*} (U), f_{!} (V)]_{Y} \subset f_{!} ([U, V]_{X})$

indexed by $U, V \in P (X)$ .

Proof of Proposition 2.4.7.1.3.

Item 1: Functoriality

Since

P (X)

is posetal, it suffices to prove Item (a), Item (b), and Item (c).

1. Proof of Item (a): We have
$\begin{aligned} [A, V]_{X} & \overset{def}{=} A^{c} \cup V \\ \subset U^{c} \cup V \\ \overset{def}{=} [U, V]_{X}, \end{aligned}$

where we have used:
1. (a) Item 1 of Proposition 2.3.11.1.2, which states that if $U \subset A$ , then $A^{c} \subset U^{c}$ .
2. (b) Item (a) of Item 1 of Proposition 2.3.11.1.2, which states that if $A^{c} \subset U^{c}$ , then $A^{c} \cup K \subset U^{c} \cup K$ for any $K \in P (X)$ .
2. Proof of Item (b): We have
$\begin{aligned} [U, V]_{X} & \overset{def}{=} U^{c} \cup V \\ \subset U^{c} \cup B \\ \overset{def}{=} [U, B]_{X}, \end{aligned}$

where we have used Item (b) of Item 1 of Proposition 2.3.11.1.2, which states that if $V \subset B$ , then $K \cup V \subset K \cup B$ for any $K \in P (X)$ .
3. Proof of Item (c): We have
$\begin{aligned} [A, V]_{X} & \subset [U, V]_{X} \\ \subset [U, B]_{X}, \end{aligned}$

where we have used Item (a) and Item (b).

This finishes the proof.

Item 2: Adjointness

This is a repetition of Item 2 of Proposition 2.3.9.1.2 and is proved there.

Item 3: Interaction With the Empty Set I

We have

\begin{aligned} [U, Ø]_{X} & \overset{def}{=} U^{c} \cup Ø \\ = U^{c}, \end{aligned}

where we have used Item 3 of Proposition 2.3.8.1.2, and we have

\begin{aligned} [Ø, V]_{X} & \overset{def}{=} Ø^{c} \cup V \\ \overset{def}{=} (X ∖ Ø) \cup V \\ = X \cup V \\ = X, \end{aligned}

where we have used:

1. Item 12 of Proposition 2.3.10.1.2 for the first equality.
2. Item 5 of Proposition 2.3.8.1.2 for the last equality.

Since $P (X)$ is posetal, naturality is automatic ().

Item 4: Interaction With

X

We have

\begin{aligned} [U, X]_{X} & \overset{def}{=} U^{c} \cup X \\ = X, \end{aligned}

where we have used Item 5 of Proposition 2.3.8.1.2, and we have

\begin{aligned} [X, V]_{X} & \overset{def}{=} X^{c} \cup V \\ \overset{def}{=} (X ∖ X) \cup V \\ = Ø \cup V \\ = V, \end{aligned}

where we have used Item 3 of Proposition 2.3.8.1.2 for the last equality. Since $P (X)$ is posetal, naturality is automatic ().

Item 5: Interaction With the Empty Set II

We have

\begin{aligned} D_{X} (D_{X} (U)) & \overset{def}{=} [[U, Ø]_{X}, Ø]_{X} \\ = [U^{c}, Ø]_{X} \\ = (U^{c})^{c} \\ = U, \end{aligned}

where we have used:

1. Item 3 for the second and third equalities.
2. Item 3 of Proposition 2.3.11.1.2 for the fourth equality.

Since $P (X)$ is posetal, naturality is automatic (), and thus we have

[[-, Ø]_{X}, Ø]_{X} ≅ {id}_{P (X)}

This finishes the proof.

Item 6: Interaction With the Empty Set III

Since

D_{X} = (-)^{c}

, this is essentially a repetition of the corresponding results for

(-)^{c}

, namely Item 5, Item 6, and Item 7 of Proposition 2.3.11.1.2.

Item 7: Interaction With Unions of Families of Subsets I

By Item 3 of Proposition 2.4.7.1.3, we have

\begin{aligned} [U, Ø]_{P (X)} & = U^{c}, \\ [U, Ø]_{X} & = U^{c} . \end{aligned}

With this, the counterexample given in the proof of Item 10 of Proposition 2.3.6.1.2 then applies.

Item 8: Interaction With Unions of Families of Subsets II

We have

\begin{aligned} [⋃_{U \in U} U, V]_{X} & \overset{def}{=} (⋃_{U \in U} U)^{c} \cup V \\ = (⋂_{U \in U} U^{c}) \cup V \\ = ⋂_{U \in U} (U^{c} \cup V) \\ \overset{def}{=} ⋂_{U \in U} [U, V]_{X}, \end{aligned}

where we have used:

1. Item 11 of Proposition 2.3.6.1.2 for the second equality.
2. Item 6 of Proposition 2.3.7.1.2 for the third equality.

This finishes the proof.

Item 9: Interaction With Unions of Families of Subsets III

We have

\begin{aligned} ⋃_{V \in V} [U, V]_{X} & \overset{def}{=} ⋃_{V \in V} (U^{c} \cup V) \\ = U^{c} \cup (⋃_{V \in V} V) \\ \overset{def}{=} [U, ⋃_{V \in V} V]_{X} . \end{aligned}

where we have used Item 6. This finishes the proof.

Item 10: Interaction With Intersections of Families of Subsets I

Let

X = {0, 1}

, let

U = {{0, 1}}

, and let

V = {{0}, {0, 1}}

. We have

\begin{aligned} ⋂_{W \in [U, V]_{P (X)}} W & = ⋂_{W \in P (X)} W \\ = {0, 1}, \end{aligned}

whereas

\begin{aligned} [⋂_{U \in U} U, ⋂_{V \in V} V]_{X} & = [{0, 1}, {0}] \\ = {0}, \end{aligned}

Thus we have

⋂_{W \in [U, V]_{P (X)}} W = {0, 1} \neq {0} = [⋂_{U \in U} U, ⋂_{V \in V} V]_{X} .

This finishes the proof.

Item 11: Interaction With Intersections of Families of Subsets II

We have

\begin{aligned} [⋂_{U \in U} U, V]_{X} & \overset{def}{=} (⋂_{U \in U} U)^{c} \cup V \\ = (⋃_{U \in U} U^{c}) \cup V \\ = ⋃_{U \in U} (U^{c} \cup V) \\ \overset{def}{=} ⋃_{U \in U} [U, V]_{X}, \end{aligned}

where we have used:

1. Item 12 of Proposition 2.3.6.1.2 for the second equality.
2. Item 6 of Proposition 2.3.7.1.2 for the third equality.

This finishes the proof.

Item 12: Interaction With Intersections of Families of Subsets III

We have

\begin{aligned} ⋂_{V \in V} [U, V]_{X} & \overset{def}{=} ⋂_{V \in V} (U^{c} \cup V) \\ = U^{c} \cup (⋂_{V \in V} V) \\ \overset{def}{=} [U, ⋂_{V \in V} V]_{X} . \end{aligned}

where we have used Item 6. This finishes the proof.

Item 13: Interaction With Binary Unions

We have

\begin{aligned} [U \cap V, W]_{X} & \overset{def}{=} (U \cap V)^{c} \cup W \\ = (U^{c} \cup V^{c}) \cup W \\ = (U^{c} \cup V^{c}) \cup (W \cup W) \\ = (U^{c} \cup W) \cup (V^{c} \cup W) \\ \overset{def}{=} [U, W]_{X} \cup [V, W]_{X}, \end{aligned}

where we have used:

1. Item 2 of Proposition 2.3.11.1.2 for the second equality.
2. Item 8 of Proposition 2.3.8.1.2 for the third equality.
3. Several applications of Item 2 and Item 4 of Proposition 2.3.8.1.2 and for the fourth equality.

For the second equality in the statement, we have

\begin{aligned} [U, V \cap W]_{X} & \overset{def}{=} U^{c} \cup (V \cap W) \\ = (U^{c} \cup V) \cap (U^{c} \cap W) \\ \overset{def}{=} [U, V]_{X} \cap [U, W]_{X}, \end{aligned}

where we have used Item 6 of Proposition 2.3.8.1.2 for the second equality.

Item 14: Interaction With Binary Intersections

We have

\begin{aligned} [U \cup V, W]_{X} & \overset{def}{=} (U \cup V)^{c} \cup W \\ = (U^{c} \cap V^{c}) \cup W \\ = (U^{c} \cup W) \cap (V^{c} \cup W) \\ \overset{def}{=} [U, W]_{X} \cap [V, W]_{X}, \end{aligned}

where we have used:

1. Item 2 of Proposition 2.3.11.1.2 for the second equality.
2. Item 6 of Proposition 2.3.8.1.2 for the third equality.

Now, for the second equality in the statement, we have

\begin{aligned} [U, V \cup W]_{X} & \overset{def}{=} U^{c} \cup (V \cup W) \\ = (U^{c} \cup U^{c}) \cup (V \cup W) \\ = (U^{c} \cup V) \cup (U^{c} \cup W) \\ \overset{def}{=} [U, V]_{X} \cup [U, W]_{X}, \end{aligned}

where we have used:

3. Item 8 of Proposition 2.3.8.1.2 for the second equality.
4. Several applications of Item 2 and Item 4 of Proposition 2.3.8.1.2 and for the third equality.

This finishes the proof.

Item 15: Interaction With Differences

We have

\begin{aligned} [U ∖ V, W]_{X} & \overset{def}{=} (U ∖ V)^{c} \cup W \\ \overset{def}{=} (X ∖ (U ∖ V)) \cup W \\ = ((X \cap V) \cup (X ∖ U)) \cup W \\ = (V \cup (X ∖ U)) \cup W \\ \overset{def}{=} (V \cup U^{c}) \cup W \\ = (V \cup (U^{c} \cup U^{c})) \cup W \\ = (U^{c} \cup W) \cup (U^{c} \cup V) \\ \overset{def}{=} [U, W]_{X} \cup [U, V]_{X}, \end{aligned}

where we have used:

1. Item 10 of Proposition 2.3.10.1.2 for the third equality.
2. Item 4 of Proposition 2.3.9.1.2 for the fourth equality.
3. Item 8 of Proposition 2.3.8.1.2 for the sixth equality.
4. Several applications of Item 2 and Item 4 of Proposition 2.3.8.1.2 and for the seventh equality.

We also have

\begin{aligned} [U ∖ V, W]_{X} & \overset{def}{=} (U ∖ V)^{c} \cup W \\ \overset{def}{=} (X ∖ (U ∖ V)) \cup W \\ = ((X \cap V) \cup (X ∖ U)) \cup W \\ = (V \cup (X ∖ U)) \cup W \\ \overset{def}{=} (V \cup U^{c}) \cup W \\ = (V \cup U^{c}) \cup (W \cup W) \\ = (U^{c} \cup W) \cup (V \cup W) \\ = (U^{c} \cup W) \cup ((V^{c})^{c} \cup W) \\ \overset{def}{=} [U, W]_{X} \cup [V^{c}, W]_{X}, \end{aligned}

where we have used:

5. Item 10 of Proposition 2.3.10.1.2 for the third equality.
6. Item 4 of Proposition 2.3.9.1.2 for the fourth equality.
7. Item 8 of Proposition 2.3.8.1.2 for the sixth equality.
8. Several applications of Item 2 and Item 4 of Proposition 2.3.8.1.2 and for the seventh equality.
9. Item 3 of Proposition 2.3.11.1.2 for the eighth equality.

Now, for the second equality in the statement, we have

\begin{aligned} [U, V ∖ W]_{X} & \overset{def}{=} U^{c} \cup (V ∖ W) \\ = (V ∖ W) \cup U^{c} \\ = (V \cup U^{c}) ∖ (W ∖ U^{c}) \\ \overset{def}{=} (V \cup U^{c}) ∖ (W ∖ (X ∖ U)) \\ = (V \cup U^{c}) ∖ ((W \cap U) \cup (W ∖ X)) \\ = (V \cup U^{c}) ∖ ((W \cap U) \cup Ø) \\ = (V \cup U^{c}) ∖ (W \cap U) \\ = (V \cup U^{c}) ∖ (U \cap W) \\ \overset{def}{=} [U, V]_{X} ∖ (U \cap W) \end{aligned}

where we have used:

10. Item 4 of Proposition 2.3.8.1.2 for the second equality.
11. Item 4 of Proposition 2.3.10.1.2 for the third equality.
12. Item 10 of Proposition 2.3.10.1.2 for the fifth equality.
13. Item 13 of Proposition 2.3.10.1.2 for the sixth equality.
14. Item 3 of Proposition 2.3.8.1.2 for the seventh equality.
15. Item 5 of Proposition 2.3.9.1.2 for the eighth equality.

This finishes the proof.

Item 16: Interaction With Complements

We have

\begin{aligned} [U^{c}, V]_{X} & \overset{def}{=} (U^{c})^{c} \cup V, \\ = U \cup V, \end{aligned}

where we have used Item 3 of Proposition 2.3.11.1.2. We also have

\begin{aligned} [U, V^{c}]_{X} & \overset{def}{=} U^{c} \cup V^{c} \\ = U \cap V \end{aligned}

where we have used Item 2 of Proposition 2.3.11.1.2. Finally, we have

\begin{aligned} [U, V]_{X}^{c} & = ((U ∖ V)^{c})^{c} \\ = U ∖ V, \end{aligned}

where we have used Item 2 of Proposition 2.3.11.1.2.

Item 17: Interaction With Characteristic Functions

We have

\begin{aligned} χ_{[U, V]_{P (X)}} (x) & \overset{def}{=} χ_{U^{c} \cup V} (x) \\ = max (χ_{U^{c}}, χ_{V}) \\ = max (1 - χ_{U} (\mod 2), χ_{V}), \end{aligned}

where we have used:

1. Item 10 of Proposition 2.3.8.1.2 for the second equality.
2. Item 4 of Proposition 2.3.11.1.2 for the third equality.

This finishes the proof.

Item 18: Interaction With Direct Images

This is a repetition of Item 10 of Proposition 2.6.1.1.4 and is proved there.

Item 19: Interaction With Inverse Images

This is a repetition of Item 10 of Proposition 2.6.2.1.3 and is proved there.

Item 20: Interaction With Direct Images With Compact Support

This is a repetition of Item 9 of Proposition 2.6.3.1.6 and is proved there.

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