Notions of Sobriety for Convergence Spaces Reinhold Heckmann - - PowerPoint PPT Presentation

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Notions of Sobriety for Convergence Spaces Reinhold Heckmann - - PowerPoint PPT Presentation

Aug 23, 2023 156 likes •377 views

Notions of Sobriety for Convergence Spaces Reinhold Heckmann AbsInt Angewandte Informatik GmbH EQU and CONV TOP (topological spaces) is not cartesian closed TOP can be extended to the CCCs EQU (equilogical spaces) and CONV

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Notions of Sobriety for Convergence Spaces Reinhold Heckmann AbsInt Angewandte Informatik GmbH

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EQU and CONV

  • TOP (topological spaces) is not cartesian closed
  • TOP can be extended to the CCCs

EQU (equilogical spaces) and CONV (convergence spaces)

  • Both are concrete categories with subspaces,

hence equalizers

  • Because of the presence of indiscrete spaces,

all subspaces are regular subspaces (equalizers)

  • For this talk:

Regular subspace of Y = equalizer of f, f ′ : Y → Z where Z is T0

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TOP and EQU / CONV

  • TOP is reflective full subcategory of EQU and CONV
  • Reflection is given by “induced topology”
  • Open, closed, closure, specialization preorder, . . .

refer to induced topology

  • Natural goal:

Extend notion of sobriety from TOP to EQU/CONV

  • General approach was proposed by Rosolini for EQU

(same as Paul Taylor’s approach in ASD)

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E-sobriety: Definition

  • Let Σ be Sierpinski space and ΩX = [X → Σ]
  • (Ω2,η,µ) is double exponential monad
  • Let SEX = Equalizer(ηΩ2X, Ω2ηX : Ω2X → Ω4X)
  • Ω2 f : Ω2X → Ω2Y restricts to SE f : SEX → SEY (functor)
  • ηX : X → Ω2X restricts to ηE

X : X → SEX (natural)

  • Define: X is E-sober if ηE

X : X ∼

= SEX

  • SE is intended to be the E-sobrification

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E-sobriety: Properties and Problems

  • A topological space is E-sober iff it is sober.
  • Closed under products (∏)?

(yes in TOP)

  • Closed under regular subspace? (yes in TOP)

(yes for retracts)

  • Y E-sober

?

⇒ [X → Y] E-sober? (yes for Y = Σ; ΩX is E-sober [Taylor])

  • Is SEX E-sober at all???
  • I cannot answer any of these questions

(no proofs and no counterexamples)

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E-sobriety: Underlying Problems

  • ηE

SEX = SEηE

X : SEX → S2

EX ?

  • If e : X ֒

→ Y is regular monic, is SEe : SEX → SEY monic?

  • For the regular monic εX : SEX ֒

→ Ω2X :

  • are SEεX and/or Ω2εX monic?
  • is Σ injective for the embedding εX?
  • Recall: Σ is not injective for general embeddings

in EQU/CONV

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Convergence Spaces

  • (X, ↓) where
  • X is a set (of points)
  • ↓ is a relation between ΦX (filters on X) and X
  • point filters converge: [x] ↓ x
  • if A ↓ x and A ⊆ B, then B ↓ x
  • f : (X, ↓X) → (Y, ↓Y) is
  • continuous if A ↓X x ⇒ f >A ↓Y fx
  • initial if also “⇐” holds
  • embedding if it is initial and injective
  • TOP ֒

→ CONV via A ↓ x iff N (x) ⊆ A

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Ω-Embedded Convergence Spaces

Def: X is Ω-embedded if ηX : X → Ω2X is an embedding

  • Ω-embedded ⇒ T0
  • For topological spaces: Ω-embedded ⇔ T0
  • Class of Ω-embedded spaces is closed under
  • product (∏)
  • subspace
  • exponentiation (if Y then [X → Y])
  • All ΩX and all SEX are Ω-embedded
  • E-sober ⇒ Ω-embedded

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Sobriety for Topological Spaces

  • Let (X,O) be a T0-topological space

Def: A ⊆ X is irreducible if {O ∈ O | O∩A = 0} is filtered. Def: X is sober if for every irreducible A ⊆ X there is a (unique) x ∈ X such that cl A = ↓x.

  • Try to mimic this with filters and convergence

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Coherent Sets and Approximation

  • Let X be an Ω-embedded convergence space

Def: For filter F ∈ ΦX and A ⊆ X, write F ◦

  • A if ∀B ∈ F . B∩A = 0

Def: A ⊆ X is coherent if there is a filter F such that F ◦

  • A

and F converges to all elements of A Def: A ⊆ X approximates x ∈ X from below (A ↑ x) if A ⊆ ↓x and there is a filter F such that F ◦

  • A and F ↓ x
  • A ↑ x ⇒ A is coherent

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Properties

  • directed ⇒ coherent ⇒ irreducible
  • In topological spaces:

coherent ⇔ irreducible

  • A ↑ x ⇒

cl A = ↓x ⇒

A = x

  • In topological spaces:

A ↑ x ⇔ cl A = ↓x

  • For f : X → Y continuous:
  • A coherent in X ⇒ f +A coherent in Y
  • A ↑ x in X ⇒ f +A ↑ fx in Y
  • In both cases, “⇔” holds if f is initial

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A-Closed Sets and A-Closure

Def: C ⊆ X is A-closed if S ⊆ C, S ↑ x ⇒ x ∈ C

  • Every regular subspace is A-closed
  • Arbritrary intersection; finite union

Def: clAB = least A-closed superset of B Def: SAX = subspace clAη+

X X of Ω2X

  • Ω2 f : Ω2X → Ω2Y restricts to SA f : SAX → SAY (functor)
  • ηX : X → Ω2X restricts to ηA

X : X → SAX (natural)

  • SAX ⊆ SEX

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A-Sober Convergence Spaces

For an Ω-embedded space X, the following are equivalent:

  • Every coherent set has a lub, and

all limit sets {x | F ↓ x} are closed under these lubs.

  • ∀ coherent A ⊆ X there is x ∈ X such that A ↑ x
  • ∀ coherent A ⊆ X there is x ∈ X such that cl A = ↓x
  • η+

X X is A-closed in Ω2X

  • ηA

X : X ∼

= SAX Def: X is A-sober if these statements hold.

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Properties of A-Sober I

  • For topological spaces:

A-sober ⇔ sober

  • Every Ω-embedded Hausdorff space is A-sober.
  • Every A-sober space is a dcpo,

and continuous functions are Scott continuous.

  • For every complete lattice L,

there is an A-sober convergence space Lγ such that TLγ = Lσ (induced top. is Scott top.).

  • For Johnstone’s non-sober complete lattice L:

Lγ is A-sober, but induced topology Lσ is not sober.

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Properties of A-Sober II

  • The class of A-sober spaces is closed under
  • product (∏)
  • A-closed subspace (hence regular subspace)
  • exponentiation (if Y then [X → Y])
  • All ΩX, all SEX, and all SAX are A-sober
  • E-sober ⇒ A-sober

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A-Sobrification

  • SAX is always A-sober.
  • If X is arbitrary and Y is A-sober,

then λgSAX→Y. g◦ηA

X : [SAX → Y] ∼

= [X → Y]

  • Hence for every continuous f : X → Y

there is a unique extension g: SAX → Y such that g◦ηA

X = f.

  • Corollary:

ΩηA

X : ΩSAX ∼

= ΩX (not only as frames, but also convergence structure)

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Repleteness: Definition

Def: Let X, Y, and Z be convergence spaces and h : X → Y continuous. (h | Z) has the iso property if λgY→Z. g◦h : [Y → Z] ∼ = [X → Z]

  • Note: If Y is A-sober,

then (ηA

X : X → SAX | Y) has the iso property.

Def: Z is replete if for all h : X → Y, if (h | Σ) has the iso property, then so has (h | Z).

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Repleteness: Properties

  • Σ is replete
  • The class of replete spaces is closed under
  • product (∏)
  • regular subspace
  • exponentiation (if Y then [X → Y])
  • Hence all ΩX and all SEX are replete
  • Hence:

E-sober ⇒ replete

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Repleteness and A-Sobriety

Let Z be replete. (ηA

Z : Z → SAZ | Σ) has the iso property

⇒ (ηA

Z : Z → SAZ | Z) has the iso property

⇒ for every continuous f : Z → Z there is a continuous g: SAZ → Z such that g◦ηA

Z = f

⇒ there is a continuous r: SAZ → Z such that r ◦ηA

Z = IdZ

⇒ Z is retract of SAZ ⇒ Z is A-sober

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Summary

Theorem: E-sober ⇒ replete ⇒ A-sober What about the opposite implications? Corollary: replete ⇒ Ω-embedded Corollary: For topological spaces X: X is replete in CONV ⇔ X is sober

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