Direct Limit Models

ofSY and the diagram

Exp^c_κ(VX) ^Exp

c

κ(Vf) //Exp^c_κ(VY)

SX

SΣ^X

OO

Sf //SY

SΣ^Y

OO

commutes. We will only use the notation Vf where it is important to explicitly distinguish between the mapsVf, Sf, Af and the morphismf, and otherwise simply write f – it is the same object, after all.

We define theexponential functor:Ex^∗ → Ex^∗, whereVX= Exp^c_κ(X)and:

AX = AX SX = Exp^c_κ(VX) Af = Af Sf = Exp^c_κ(Vf) AΣ^X = id_AX SΣ^X = Exp^c_κ(VΣ^X).

Vfis in fact continuous wheneverf:X →Yis a morphism, becauseVf Exp^c_κ(VX) =Sf is the exponential of a continuous map andVf◦VΣ^X = VΣ^Y◦Vf, which is also continuous.

Σ^X is a morphism Σ^X : X → X and the following diagram commutes, which means that X 7→ Σ^Xis a natural transformation from the identity to:

X ^{f //}Y X

Σ^X

OO

f //Y

Σ^Y

OO

Theextension maps categoryEx is the full⁶ subcategory given by those objectsXwhose uni-verse spacesVXareκ-compact Hausdorff. And thecategory of (extension map) hyperuniverses⁷ EHypis the full subcategory ofExgiven by those objectsXwhereVΣ^Xis bijective and thus a homeomorphism. IfVXisκ-compact Hausdorff, thenVXalso is, so the restriction Exis a functor fromExtoEx.

We define the κ-Cech-Stone compactification functorˇ β from Ex^∗ to Ex by VβX = βVX and SβX = cl(ι_X[SX]) = βSX, and using the universal property of theκ-ˇCech-Stone compactifi-cation of a topological space, let SΣ^βX : SβX → Exp(VβX)be the unique map such that the diagram

Exp^c_κ(VX) ^Exp

c

κ(ι_VX) //Exp_κ(VβX)

SX

SΣ^X

OO

ι_SX //SβX

SΣ^X

OO

commutes, that is, such thatVιX = ι_VX defines a morphismι_X : X → βX. For a morphism

6That means it containsallmorphismsf:X→YofEx^∗whenever it contains the objectsXandY.

7Note that for technical reasons we include the trivial object with only one point here, although we do not consider it a proper hyperuniverse.

2.5. DIRECT LIMIT MODELS 63

f:X→ Y, letVβf= βVf. Then all cells in the diagram

Exp_κ(VβX) ^{Expκ(Vβf) //}Exp_κ(VβY)

Exp^c_κ(VX) ^Exp

cκ(Vf) //

Exp^c_κ(VιX)

ii

Exp^c_κ(VY)

Exp^c_κ(VιY)

55

SX

SΣ^X

OO

Sf //

SιX

uu

SY

SΣ^Y

OO

SιY

))SβX

SΣ^X

OO

Sβf //SβY

SΣ^Y

OO

commute, so SΣ^βY◦Sβf◦SιX = Exp_κ(Vβf)◦SΣ^βX◦SιX. Since SιX is an epimorphism (its image is dense), this impliesSΣ^βY◦Sβf = Exp_κ(Vβf)◦SΣ^βX and hence βfis in fact a mor-phism. The map X 7→ ι_X is a natural transformation from the identity to βand β is a left adjoint functor. Its right adjoint is the inclusion fromExtoEx^∗. Thus by the following lemma, direct limits inExexist and they are theκ-ˇCech-Stone compactifications of the corresponding direct limits inEx^∗:

Lemma 46. LethX_i,f_ijibe a directed system inEx^∗. Then the direct limithX,f_iiexists:

hVX,Vfii = lim

→

TophVX_i,Vfiji , hSX,Sfii = lim

→

TophSX_i,Sfiji andSΣ^X is the unique map such that for alli,SΣ^X◦Sf_i = Exp^c_κ(Vf_i)◦SΣ^Xi.

Proof. The definition ofSΣ^Xdoes make sense because indeed

Exp^c_κ(Vf_j)◦SΣ^X_j ◦Sf_ij = Exp^c_κ(Vf_j)◦Exp^c_κ(Vf_ij)◦SΣ^X_i = Exp^c_κ(Vf_i)◦SΣ^X_i

wheneveri < j, so the direct limit property ofSXapplies.

We show that the objectXwith the morphismsf_idefined by those topological limits is in fact a direct limit of the given system inEx^∗.

For eachi, leth_i :X_i →Zbe a morphism and assume that for allj >i,h_i= h_j◦f_ij. We then defineVh :VX→ VZusing the direct limit property ofVX, which is the unique candidate for a suitable morphismh : X → Z, and we just have to show that it is in fact a morphism. For

eachi, every path fromSX_ito Exp^c_κ(VZ)is equivalent in the diagram

Exp^c_κ(VX) ^Exp

c

κ(Vh) //Exp^c_κ(VZ)

Exp^c_κ(VXi)

Exp^cκ(Vh_i) 44

Exp^cκ(Vf_i)

jj

SXi SΣ^Xi

OO

Sfi

ss ^Shi ++

SX

SΣ^X

OO

Sh //SZ

SΣ^Z

OO

and in particularSΣ^Z◦Sh◦Sf_i = Exp^c_κ(Vh)◦SΣ^X◦Sf_i. SincehSX,Sf_iiis the direct limit of theSX_i, it follows thatSΣ^Z◦Sh = Exp^c_κ(Vh)◦SΣ^X.

We recursively define functors ^β from Ex to Ex and morphisms Σ^X_α,β : ^αX → ^βX for ordinalsα6β:

⁰X = X Σ^X_α,α = id^α_X ⁰f = f ^α+1X = ^αX Σ^X_α,β+1 = Σ^βX◦Σ^X_α,β ^α+1f = ^αf For limit ordinalsγ, let

h^γX,Σ^X_α,γi_α<γ = lim

→

Exh^αX,Σ^X_α,βi_α6β<γ

and iff : X → Y is a morphism, we use the direct limit property of^γX: ^γfis defined as the unique morphism such that^γf◦Σ^X_α,γ = Σ^Y_α,γ◦^αffor allα < γ. ThenX 7→ Σ^X_α,β is a natural transformation from^αto^β, that is, for every morphismf : X→ Y, the following diagram commutes:

^αX ^Σ

X

α,β //

^αf

^βX

^βf

^αY ^Σ

Y

α,β //^βY

Lemma 47. For each objectXin Ex, SΣ^X_κ,κ+1 is surjective and as a consequence, all SΣ^X_β,α withκ6 β6αare.

Proof. The unionD= S

α<κrng VΣ^X_α,κ

is dense inV^κX= βD, so by Lemma 22, the set of κ-small subsets ofDis dense in Exp^c_κ(V^κX) =S^κ+1X. But for eachκ-smallS ⊆ D, there exists anα < κand aκ-smallSe⊆V^αX, such thatS =VΣ^X_α,κ[eS]. Therefore,

[

α<κ

rng Exp^c_κ(VΣ^X_α,κ)

= [

α<κ

rng SΣ^X_κ,κ+1◦SΣ^X_α+1,κ

⊆rng SΣ^X_κ,κ+1

is dense inS^κ+1X. ButSΣ^X_κ,κ+1is continuous and its domain isκ-compact, hence its image is closed and therefore rng

SΣ^X_κ,κ+1

= S^κ+1X.

2.5. DIRECT LIMIT MODELS 65

The equivalence relationsE_α ⊆(V^κX)²forα >κdefined by xE_αy ⇔ VΣ^X_κ,α(x) = VΣ^X_κ,α(y)

are monotonously increasing and thus the sequence has to be eventually constant, that is, there is someαsuch thatVΣ^αX is a homeomorphism, so^αXis an object ofEHyp. Letγ_X be the least suchα.

For objectsXandYofEx, morphismsf :X→ Yandα= max{γ_X,γ_Y}, we define:

^∞X = ^γXX ^∞f = Σ^Y_γ

Y,α

−1

◦^αf◦Σ^X_γ

X,α

Σ^X_β,∞ =



 Σ^X_β,γ

X forβ6γ_X

Σ^X_γ

X,β

−1

forβ > γ_X

^∞is a functor fromExtoEHypandX7→ Σ^X_β,∞is a natural transformation from^βto^∞. V^∞Xis a quotient ofV^κX. Next we show thatΣ^X_0,∞is initial among the morphisms fromX to objects ofEHyp:

Theorem 48. LetXbe an object of Ex,Y an object ofEHyp andf : X → Y. Then there is a unique morphismg :^∞X→ Y such thatf= g◦Σ^X_0,∞, namelyg =^∞f.

X

Σ^X_0,∞ ''

f //Y

^∞X

g

77

The functor^∞ :Ex → EHypis a left adjoint of the inclusion functor.

Therefore, for every objectXofEx^∗ andYofEHypand every morphismf :X→ Y, there is a unique morphismg :^∞βX→ Y withf= g◦Σ^βX_0,∞◦ι_X, namelyg = (ι_Y)⁻¹◦^∞βf.

Proof. Sinceγ_Y = 0 andΣ_0,γX is a natural transformation, ^∞f◦Σ^X_0,∞ = Σ^Y_γ

Y,γX

−1

◦^γXf◦Σ^X_γ

X,γX◦Σ^X_0,γ

X = Σ^Y_0,γ

X

−1

◦^γXf◦Σ^X_0,γ

X = f, so^∞fhas the required property.

It remains to show that for every morphismg, the equationf = g◦Σ^X_0,∞ implies g = ^∞f.

By induction onα6 γ_X, we will show that the following diagram commutes:

^αX

^αf

Σ^X_α,∞

//^∞X

g

^αY ^Σ Y

Y0,α

oo

Then the caseα =γ_Xwill imply our claim, proving uniqueness:

^∞f = Σ^Y_0,γ

X

−1

◦^γXf = g◦Σ^X_γ

X,∞ = g The caseα= 0 is just our assumptionf= g◦Σ^X_0,∞.

Now letα = β+1. Applying to the induction hypothesis forβ, we obtain the left cell in the diagram

^αX

^αf

Σ^X_α,γX+1

//^γX+1X

g

^γXX

Σ^X_γX,γX+1

oo

g

^αY ^Σ Y

Y

oo 1,α ^Σ Y

Y

oo 0,1

and the right cell commutes becauseΣis a natural transformation. SinceΣ^X_γ

X,γX+1 = Σ^∞X is an isomorphism, this proves the caseα.

Finally, letαbe a limit ordinal and assume the induction hypothesis for allβ < α. Then every cell in the diagram

^αX

^αf

Σ^X_α,∞

//^∞X

g

^βX

Σ^X_β,α

jj

Σ^X_β,∞

44

^βf

^βY

Σ^Y_β,α

tt

Σ^Y_0,β

**^αY Y^Σ

Y0,α

oo

commutes and hence Σ^Y_0,α−1

◦^αf◦Σ^X_β,α = g◦Σ^X_α,∞◦Σ^X_β,α : ^βX →Y for all β < α. So by the direct limit property of ^αX, it follows that

Σ^Y_0,α −1

◦^αf = g◦Σ^X_α,∞.

To prove that^∞is a left adjoint, we prove that

Φ_X,Y :hom(^∞X,Y)→ hom(X,Y), f7→ f◦Σ^X_0,∞

has the required property:

Φ

X,eYe(h₁◦f◦^∞h₀) =h₁◦f◦^∞h₀◦Σ^X_0,^e_∞ = h₁◦f◦Σ^X_0,∞◦h₀ = h₁◦Φ_X,Y(f)◦h₀, for all morphismsh₀ :Xe → X,h₁ :Y → Yeandf:X →Y.

Let X be an object in Ex^∗ and A ⊆ VX open. A morphism f : X → Y is called an A-homeomorphismif

2.5. DIRECT LIMIT MODELS 67

• f(x) 6= f(a)for everyx∈VXanda ∈Awithx6=a, and

• f[U]is open inVY for every openU⊆A.

Then ifB⊆Ais open,fis also aB-homeomorphism. And whenevergis anf[A]-homeomorphism, g◦fis anA-homeomorphism.

Ais calledtransitiveif

Σ^X[A∩SX] ⊆ A,

and it is calledpersistent if it is transitive, Hausdorff, open, locallyκ-compact and Σ^X is an A-homeomorphism. Intuitively,Ais transitive iff all the elements of the classAin the model Xare subsets ofAagain, and as we will see the notion of persistence is chosen such that a persistent set is protected from being collapsed in the construction process of the ^αX, so that it – or rather its isomorphic image – is still present in the hyperuniverse^∞X.

Lemma 49. LetXbe an object ofEx^∗ andA⊆VX.

1. IfAis transitive inX,f[A]is transitive inY for everyf:X→ Y.

ι_X[A], allΣ^X_0,α[A]andA∪(A∩AX)are transitive inβX,^α[X]andX.

2. IfAis persistent andf :X → Y is anA-homeomorphism, thenfis a A∪(A∩ AX)-homeomorphism. If moreoverf[SX]is dense inSY, thenf[A]is persistent. In particular, Σ^X[A]andA∪(A∩AX)are persistent inXandι_X[A]is persistent inβX.

3. IfhX_α,f_α,βi_α,β<λis a directed system inEx^∗, eachf_α,βis anf_0,α[A]-homeomorphism andf_0,α[A]is persistent for eachα < λ, then if

hX_λ,f_α,λi_α<λ = lim

→

Ex^∗hX_α,f_α,βi_α,β<λ

is the direct limit, everyf_α,λis anf_0,α[A]-homeomorphism andf_0,λ[A]is persistent.

Proof. (1): These claims are easily verified by direct calculation:

Σ^Y[f[A]∩SY] = Σ^Y[f[A∩SX]] =Exp^c_κ(f)[Σ^X[A∩SX]]

⊆ Exp^c_κ(f)[A] = (f[A])

Σ^X[(A∪(A∩AX))∩SX] = Σ^X[A] = Exp^c_κ(Σ^X)[A] =(Σ^X[A])

= (Σ^X[A∩SX]∪Σ^X[A∩AX])⊆(A∪(A∩AX)) (2): Sincefis the union of Exp^c_κ(Vf)andAf, its injectivity onAandA∩AAfollows from the injectivity offon A. It is also open on A∩AA, because Afis. To prove its openness on Ait suffices to check the images of subbase sets of the formV and♦Vfor openV ⊆ A:

Σ^X[V] = Exp^c_κ(Σ^X)[V] = (Σ^X[V])and Σ^X[♦V] = Exp^c_κ(Σ^X)[♦V] = ♦(Σ^X[V]).

It remains to show that nob /∈ A∪(A∩AX)is mapped into the image of that set. But such a bis either an atom – then f(b) = f(b) ∈/ f(A∩AX) = f(A∩AX) –, or it contains an

elementz /∈ A. In that case,f(z) ∈/ f[A]and thereforef(b)contains an element not inf[A], sof(b) ∈/ (f[A]) = f[A].

We have already seen that f[A] is transitive. As the homeomorphic image of a Hausdorff, locally κ-compact set, it also has these properties. Since Σ^X[A] ⊆ A∪(A∩AX)and fis anA∪(A∩AX)-homeomorphism,f◦Σ^X =Σ^Y◦fis anA-homeomorphism. In particular, Σ^Y mapsf[A]homeomorphically onto its image, which is open.

Now assume thatf[SX]is dense inSY. It remains to prove that y ∈ f[A]wheneverΣ^Y(y) ∈ Σ^Y[f[A]]. So assume Σ^Y(y) = Σ^Y(f(x)) for some x ∈ A. Then x has a κ-compact neigh-borhood U in A, and f[U] 3 f(x) as well as V = Σ^Y[f[U]] 3 Σ^Y(y) are κ-compact neigh-borhoods. Thus if y /∈ f[A], then in particular y /∈ f[A]∩(Σ^Y)⁻¹[V] = f[U], and hence y has a neighborhood W ⊆ (Σ^Y)⁻¹[V] disjoint from f[U]and therefore disjoint from A. But f[SX]∩W = f[SX\A]∩Wis dense inW. So

Σ^Y(y) ∈ Σ^Y[cl(f[SX\A])] ⊆ cl(Σ^Y[f[SX\A]]) ⊆ cl(f[Σ^X[SX\A]]).

But since f◦Σ^X is an A-homeomorphism, f[Σ^X[SX\A]] is disjoint from the open set f[Σ^X[A]] =Σ^Y[f[A]], contradictingΣ^Y(y) ∈Σ^Y[f[A]].

We have already seen that A∪(A∩AX) is transitive andΣ^X = Σ^X is aA∪(A∩ AX)-homeomorphism. To prove thatA∪(A∩AX)is persistent, we only have to verify that it is open, locallyκ-compact and Hausdorff. A∩AXis an open subset ofA, so for that part of the union these properties are immediate. A is open becauseAis, and Hausdorff and locally κ-compact by Lemma 23.

As a transitive open subset of A∪(A∩AX), Σ^X[A] is persistent, too. Finally, ι_X is an A-homeomorphism with a dense image, soι_X[A]is persistent.

(3): Let us begin by showing that f_0,λ is anA-homeomorphism (and analogously everyf_α,λ is anf_0,α[A]-homeomorphism). Ifx∈VX₀andy∈ Aare distinct, thenf_0,α(x)6= f_0,α(y)for everyα, sof_0,λ(x) 6=f_0,λ(y). In particular,f_0,λ is injective onA. Thus ifU⊆Ais open, then since everyf⁻¹_α,λ[f_0,λ[U]] =f_0,α[U]is open,f_0,λ[U]is open, too.

As in (2), it follows without any density assumption that f_0,λ[A]is open, Hausdorff, locally κ-compact andΣ^Xλ mapsf_0,λ[A]homeomorphically onto its image, which is open. It remains to show that noex /∈f_0,λ[A]is mapped intoΣ^Xλ[f_0,λ[A]].

ex = f_α,λ(x) for some x ∈ VX_α. Let VY = f_α,λ[X_α]. Since Σ^Xλ◦f_α,λ = f_α,λ◦Σ^X, the restriction SΣ^Y = SΣ^Xλ SY is a map into Exp^c_κ(f_α,λ[X_α]) = Exp^c_κ(VY). Thusf_α,λ also is an A-homeomorphism from X_α to Y, and then it is surjective, so (2) applies. Hence A is persistent inY, which proves thatΣ^Xλ(ex) = Σ^Y(f_α,λ(x)) ∈/ Σ^Y[f_0,λ[A]] =Σ^Xλ[f_0,λ[A]].

Theorem 50. IfXis an object ofEx^∗andA⊆VXis persistent, thenΣ^βX_0,α◦ι_X[A]is persistent in^αβX, andΣ^βX_0,α◦ι_Xis anA-homeomorphism for every ordinalα.

In particular,Σ^βX_0,∞◦ι_X[A]is persistent in the hyperuniverse^∞βXand homeomorphic toA.

Proof. The proof is by induction onαand follows immediately from Lemma 49: First of all,ι_X

Im Dokument Topological set theories and hyperuniverses (Seite 67-75)