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

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 (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 T 0 2

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) 3

E-sobriety: Definition • Let Σ be Sierpinski space and Ω X = [ X → Σ ] ( Ω 2 , η , µ ) is double exponential monad • Let S E X = Equalizer ( η Ω 2 X , Ω 2 η X : Ω 2 X → Ω 4 X ) • Ω 2 f : Ω 2 X → Ω 2 Y restricts to S E f : S E X → S E Y (functor) • η X : X → Ω 2 X restricts to η E • X : X → S E X (natural) X : X ∼ Define: X is E-sober if η E • = S E X • S E is intended to be the E-sobrification 4

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 ; Ω X is E-sober [Taylor]) Is S E X E-sober at all??? • • I cannot answer any of these questions (no proofs and no counterexamples) 5

E-sobriety: Underlying Problems η E S E X = S E η E X : S E X → S 2 • E X ? • If e : X ֒ → Y is regular monic, is S E e : S E X → S E Y monic? → Ω 2 X : For the regular monic ε X : S E X ֒ • • are S E ε X and/or Ω 2 ε X monic? • is Σ injective for the embedding ε X ? • Recall: Σ is not injective for general embeddings in EQU/CONV 6

Convergence Spaces • ( X , ↓ ) where • X is a set (of points) • ↓ is a relation between Φ X (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 → CONV via A ↓ x iff N ( x ) ⊆ A • TOP ֒ 7

Ω -Embedded Convergence Spaces Def: X is Ω -embedded if η X : X → Ω 2 X is an embedding • Ω -embedded ⇒ T 0 • For topological spaces: Ω -embedded ⇔ T 0 • Class of Ω -embedded spaces is closed under • product ( ∏ ) • subspace • exponentiation (if Y then [ X → Y ] ) • All Ω X and all S E X are Ω -embedded • E-sober ⇒ Ω -embedded 8

Sobriety for Topological Spaces • Let ( X , O ) be a T 0 -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 9

Coherent Sets and Approximation • Let X be an Ω -embedded convergence space Def: For filter F ∈ Φ X 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 10

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 11

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: clA B = least A-closed superset of B Def: S A X = subspace clA η + X X of Ω 2 X Ω 2 f : Ω 2 X → Ω 2 Y restricts to S A f : S A X → S A Y (functor) • η X : X → Ω 2 X restricts to η A • X : X → S A X (natural) • S A X ⊆ S E X 12

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 Ω 2 X • X : X ∼ η A • = S A X Def: X is A-sober if these statements hold. 13

Properties of A-Sober I • A-sober ⇔ sober For topological spaces: • 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 L γ such that TL γ = L σ (induced top. is Scott top.). • For Johnstone’s non-sober complete lattice L : L γ is A-sober, but induced topology L σ is not sober. 14

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 Ω X , all S E X , and all S A X are A-sober • • E-sober ⇒ A-sober 15

A-Sobrification • S A X is always A-sober. • If X is arbitrary and Y is A-sober, X : [ S A X → Y ] ∼ then λ g S A X → Y . g ◦ η A = [ X → Y ] • Hence for every continuous f : X → Y there is a unique extension g : S A X → Y such that g ◦ η A X = f . X : Ω S A X ∼ Ωη A • Corollary: = Ω X (not only as frames, but also convergence structure) 16

Repleteness: Definition Def: Let X , Y , and Z be convergence spaces and h : X → Y continuous. ( h | Z ) has the iso property if λ g Y → Z . g ◦ h : [ Y → Z ] ∼ = [ X → Z ] • Note: If Y is A-sober, then ( η A X : X → S A X | 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 ) . 17

Repleteness: Properties • Σ is replete • The class of replete spaces is closed under • product ( ∏ ) • regular subspace • exponentiation (if Y then [ X → Y ] ) • Hence all Ω X and all S E X are replete • Hence: E-sober ⇒ replete 18

Repleteness and A-Sobriety Let Z be replete. ( η A Z : Z → S A Z | Σ ) has the iso property ⇒ ( η A Z : Z → S A Z | Z ) has the iso property ⇒ for every continuous f : Z → Z there is a continuous g : S A Z → Z such that g ◦ η A Z = f ⇒ there is a continuous r : S A Z → Z such that r ◦ η A Z = Id Z ⇒ Z is retract of S A Z ⇒ Z is A-sober 19

Summary E-sober ⇒ replete ⇒ A-sober Theorem: What about the opposite implications? Corollary: replete ⇒ Ω -embedded Corollary: For topological spaces X : X is replete in CONV ⇔ X is sober 20