Let X be a set.

  1. 1. Functoriality. The assignment UUUU defines a functor
    :(P(P(X)),)(P(X),).

    In particular, for each U,VP(P(X)), the following condition is satisfied:

    • If UV, then VVVUUU.

  2. 2. Oplax Associativity. We have a natural transformation

    with components

    AA(UAU)UAAAU

    for each AP(P(P(X))).

  3. 3. Left Unitality. The diagram

    commutes, i.e. we have

    V{U}V=U.

    for each UP(X).

  4. 4. Oplax Right Unitality. The diagram

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

    {x}χX(U){x}U

    in general, where UP(X). However, when U is nonempty, we have

    {x}χX(U){x}U.
  5. 5. Interaction With Unions I. The diagram

    commutes, i.e. we have

    WUVW=(UUU)(VVV)

    for each U,VP(P(X)).

  6. 6. Interaction With Unions II. The diagram
    commute, i.e. we have
    U(VVV)=VV(UV),(UUU)V=UU(UV)

    for each U,VP(P(X)) and each U,VP(X).

  7. 7. Interaction With Intersections I. We have a natural transformation

    with components

    (UUU)(VVV)WUVW

    for each U,VP(P(X)).

  8. 8. Interaction With Intersections II. The diagrams
    commute, i.e. we have
    U(VVV)=VV(UV),(UUU)V=UU(UV)

    for each U,VP(P(X)) and each U,VP(X).

  9. 9. Interaction With Differences. The diagram

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

    WUVW(UUU)(VVV)

    in general, where U,VP(P(X)).

  10. 10. Interaction With Complements I. The diagram

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

    WUcWUUUc

    in general, where UP(P(X)).

  11. 11. Interaction With Complements II. The diagram

    commutes, i.e. we have

    (UUU)c=UUUc

    for each UP(P(X)).

  12. 12. Interaction With Complements III. The diagram

    commutes, i.e. we have

    (UUU)c=UUUc

    for each UP(P(X)).

  13. 13. Interaction With Symmetric Differences. The diagram

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

    WUVW(UUU)(VVV)

    in general, where U,VP(P(X)).

  14. 14. Interaction With Internal Homs I. The diagram

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

    W[U,V]P(X)W[UUU,VVV]X

    in general, where UP(P(X)).

  15. 15. Interaction With Internal Homs II. The diagram

    commutes, i.e. we have

    [UUU,V]X=UU[U,V]X

    for each UP(P(X)) and each VP(X).

  16. 16. Interaction With Internal Homs III. The diagram

    commutes, i.e. we have

    [U,VVV]X=VV[U,V]X

    for each UP(X) and each VP(P(X)).

  17. 17. Interaction With Direct Images. Let f:XY be a map of sets. The diagram

    commutes, i.e. we have

    UUf(U)=Vf(U)V

    for each UP(X), where f(U)=def(f)(U).

  18. 18. Interaction With Inverse Images. Let f:XY be a map of sets. The diagram

    commutes, i.e. we have

    VVf1(V)=Uf1(U)U

    for each VP(Y), where f1(V)=def(f1)1(V).

  19. 19. Interaction With Direct Images With Compact Support. Let f:XY be a map of sets. The diagram

    commutes, i.e. we have

    UUf!(U)=Vf!(U)V

    for each UP(X), where f!(U)=def(f!)!(U).

  20. 20. Interaction With Unions of Families I. The diagram

    commutes, i.e. we have

    UAAAU=AA(UAU)

    for each AP(P(X)).

  21. 21. Interaction With Unions of Families II. Let X be a set and consider the compositions

    given by

    AUAAAU,AAA(UAU),AUAAAU,AAA(UAU)

    for each AP(P(P(X))). We have the following inclusions:

    All other possible inclusions fail to hold in general.

Item 1: Functoriality
Since P(X) is posetal, it suffices to prove the condition (). So let U,VP(P(X)) with UV. We claim that
VVVUUU.

Indeed, if xVVV, then xV for all VV. But since UV, it follows that xU for all UU as well. Thus xUUU, which gives our desired inclusion.

Item 2: Oplax Associativity
We have
AA(UAU)=def{xX | for each AA,we have xUAU}=def{xX | for each AA and eachUA, we have xU}={xX | for each UAAA,we have xU}{xX | for each UAAA,we have xU}=defUAAAU.

Since P(X) is posetal, naturality is automatic (). This finishes the proof.

Item 3: Left Unitality
We have
V{U}V=def{xX | for each V{U},we have xU}={xX | xU}=U.

This finishes the proof.

Item 4: Oplax Right Unitality
If U=Ø, then we have
{u}χX(U){u}={u}Ø{u}=X,

so {u}χX(U){u}=XØ=U. When U is nonempty, we have two cases:

  1. 1. If U is a singleton, say U={u}, we have
    {u}χX(U){u}={u}=defU.
  2. 2. If U contains at least two elements, we have
    {u}χX(U){u}=ØU.

This finishes the proof.

Item 5: Interaction With Unions I
We have
WUVW=def{xX | for each WUV,we have xW}={xX | for each WU and eachWV, we have xW}=def{xX | for each WU,we have xW}={xX | for each WV,we have xW}=def(WUW)(WVW)=(UUU)(VVV).

This finishes the proof.

Item 6: Interaction With Unions II
Omitted.
Item 7: Interaction With Intersections I
We have

(UUU)(VVV)=def{xX | for each UU,we have xU}={xX | for each VV,we have xV}={xX | for each WUV,we have xW}{xX | for each WUV,we have xW}=defWUVW.

Since P(X) is posetal, naturality is automatic (). This finishes the proof.

Item 8: Interaction With Intersections II
Omitted.
Item 9: Interaction With Differences
Let X={0,1}, let U={{0},{0,1}}, and let V={{0}}. We have

WUVU=W{{0,1}}W={0,1},

whereas

(UUU)(VVV)={0}{0}=Ø.

Thus we have

WUVW={0,1}Ø=(UUU)(VVV).

This finishes the proof.

Item 10: Interaction With Complements I
Let X={0,1} and let U={{0}}. We have
WUcU=W{Ø,{1},{0,1}}W=Ø,

whereas

UUUc={0}c={1}.

Thus we have

WUcU=Ø{1}=UUUc.

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={0,1}, let U={{0,1}}, and let V={{0},{0,1}}. We have
WUVW=W{{0}}W={0},

whereas

(UUU)(VVV)={0,1}{0}=Ø,

Thus we have

WUVW={0}Ø=(UUU)(VVV).

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
This is a repetition of Item 4 of Proposition 2.6.3.1.6 and is proved there.
Item 20: Interaction With Unions of Families I
This is a repetition of Item 20 of Proposition 2.3.6.1.2 and is proved there.
Item 21: Interaction With Unions of Families II
This is a repetition of Item 21 of Proposition 2.3.6.1.2 and is proved there.


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