A Vietoris functor for bispaces and d-frames Tom a s Jakl (joint - - PowerPoint PPT Presentation

A Vietoris functor for bispaces and d-frames

Tom´ aˇ s Jakl

(joint work with Aleˇ s Pultr and Achim Jung)

Department of Applied Mathematics Charles University in Prague AND School of Computer Science University of Birmingham

TACL, 30 June 2017

0 / 16

J´

• nsson-Tarski duality

StoneSpop Bool

∼ =

Clp Ult For propositional logic

J´

• nsson-Tarski duality

StoneSpop Bool

∼ =

Clp Ult DGFop MA

∼ =

Uop U For modal logic

1 / 16

DGF are V-coalgebras

Let (X, τ) be a Stone space. V(X, τ)

def

≡ (KX, Vτ) where Vietoris functor

• 1. KX = compact subsets of X
• 2. Vτ is generated by ×

V , + ♦V

(for all V ∈ τ) where

× V = {K ∈ KX | K ⊆ V }

+

♦V = {K ∈ KX | K ∩ V = ∅}

Theorem (Kupke, Kurz, Venema 2003)

The category of descriptive general frames and the category of V-coalgebras are isomorphic. Continuous X → V(X)

2 / 16

Modal algebras are M-algebras

Let B be a Boolean algebra. M(B)

def

≡ BA a : a ∈ B / ≈ where ≈ is generated by (a ∧ b) ≈ a ∧ b and 1 ≈ 1

Theorem (folklore?)

The category of modal algebras and the category of M-algebras are isomorphic. Homomorphisms M(B) → B

3 / 16

New picture

StoneSpop Bool

∼ =

Clp Ult Coalg(V)op Alg(M)

∼ =

Uop U Vop M M is an “algebraic dual” of V

4 / 16

New picture

StoneSpop Bool

∼ =

Clp Ult Coalg(V)op Alg(M)

∼ =

Uop U Vop M M is an “algebraic dual” of V

4 / 16

Frames

A complete lattice (L; , ∧, 0, 1) is a frame iff b ∧ (

• i

ai) =

• i

(b ∧ ai) ( = ⇒ complete Heyting algebra)

Example

Ω(X; τ) = (τ; , ∩, ∅, X) Ω(f : X → Y ): U → f −1[U]

5 / 16

Frames

A complete lattice (L; , ∧, 0, 1) is a frame iff b ∧ (

• i

ai) =

• i

(b ∧ ai) ( = ⇒ complete Heyting algebra)

Example

Ω(X; τ) = (τ; , ∩, ∅, X) Ω(f : X → Y ): U → f −1[U] Topop Frm

⊥

Ω Σ

5 / 16

Frame M

KRegSpop KRegFrm

∼ =

Ω Σ Compact regular spaces

♦a = ¬(¬a)

Frame M

KRegSpop KRegFrm

∼ =

Ω Σ V M? Compact regular spaces M(L)

def

≡ Fr a, ♦a : a ∈ L /≈ where

a ∧ b ≈ (a ∧ b) 1 ≈ 1 ♦a ∨ ♦b ≈ ♦(a ∨ b) ♦0 ≈ 0 a ∧ ♦b ♦(a ∧ b) (a ∨ b) a ∨ ♦b

• ↑ ai ≈ (
• ↑ ai)
• ↑ ♦ai ≈ ♦(
• ↑ ai)

♦a = ¬(¬a)

6 / 16

The whole perspective

biKReg KRegSp PriesSp StoneSp Compact regular bitopological spaces Priestley spaces

The whole perspective

biKRegop KRegSpop PriesSpop StoneSpop Compact regular bitopological spaces Priestley spaces d-KReg KRegFrm DLat Bool ∼ = ∼ = ∼ = ∼ = Compact regular d-frames

7 / 16

The whole perspective

biKRegop KRegSpop PriesSpop StoneSpop Compact regular bitopological spaces Priestley spaces d-KReg KRegFrm DLat Bool ∼ = ∼ = ∼ = ∼ = Compact regular d-frames

∀X ∈ {StoneSp, KRegSp, PriesSp} ∃V: X → X ∀A ∈ {Bool, KRegFrm, DLat} ∃M: A → A and Coalg(V)op ∼ = Alg(M)

(whenever X op ∼ = A)

7 / 16

The task: generalise V’s and M’s

biKRegop d-KReg

∼ =

X op A

∼ =

I op J Vop M Wop? Md? such that I ◦ V ∼ = W ◦ I and J ◦ M ∼ = Md ◦ J

8 / 16

D-frames (Jung & Moshier, 2006)

D-frame is a structure L = (L+, L−; con, tot) where

◮ L+ and L− are frames ◮ con, tot ⊆ L+×L−

(+ axioms, e.g. (x+, x−) ∈ con, x′

+ ≤ x+, x′ − ≤ x−

(x′

+, x′ −) ∈ con

)

Example

Ωd(X, τ+, τ−) = (τ+, τ−, conX, totX) (U, V ) ∈ conX

def

≡ U ∩ V = ∅ (U, V ) ∈ totX

def

≡ U ∪ V = X

9 / 16

D-frames (Jung & Moshier, 2006)

D-frame is a structure L = (L+, L−; con, tot) where

◮ L+ and L− are frames ◮ con, tot ⊆ L+×L−

(+ axioms, e.g. (x+, x−) ∈ con, x′

+ ≤ x+, x′ − ≤ x−

(x′

+, x′ −) ∈ con

)

biTopop d-Frm

⊥

Ωd Σd

9 / 16

Example: embedding the frame duality

biKRegop d-KReg

∼ =

KRegSpop KRegFrm

∼ =

I op J

◮ I : (X, τ) → (X, τ, τ) ◮ J : L → (L, L, conL, totL) where

(a, b) ∈ conL iff a ∧ b = 0 (a, b) ∈ totL iff a ∨ b = 1

10 / 16

Example: embedding the Priestley duality

biKRegop d-KReg

∼ =

PriesSpop DLat

∼ =

I op J

◮ I : (X, τ, ≤) → (X, τ+, τ−) where

τ+ = Up(X, ≤) ∩ τ τ− = Down(X, ≤) ∩ τ

◮ J : D → (Idl(D), Filt(D), conD, totD) where

(I, F) ∈ conD iff ∀i ∈ I, f ∈ F : i ≤ f (I, F) ∈ totD iff I ∩ F = ∅

11 / 16

biKRegop d-KReg

∼ =

Ωd Σd W? Md?

12 / 16

biKRegop d-KReg

∼ =

Ωd Σd W? Md? W: (X; τ+, τ−) → (KcX; Vτ+, Vτ−) where

• 1. KcX = compact convex subsets of X

(Note: (≤τ+) = (≥τ−))

• 2. Vτ+ is generated by ×

U+, + ♦U+

(for all U+ ∈ τ+)

• 3. Vτ− is generated by ×

U−, + ♦U−

(for all U− ∈ τ−)

12 / 16

Free d-frame construction

For (B+, ≈+, B−, ≈−, con1, tot1) where

• 1. L+ = Fr B+ /≈+
• 2. L− defined similarly
• 3. con1, tot1 ⊆ B+×B−

Theorem

(L+, L−; CONcon1, TOTtot1) is a d-frame if Closure under con-operations in L+×L− Closure under tot-operations in L+×L− DCPO↑ ↓DL∨,∧ con1 ↑DL∨,∧ tot1

13 / 16

Free d-frame construction

For (B+, ≈+, B−, ≈−, con1, tot1) where

• 1. L+ = Fr B+ /≈+
• 2. L− defined similarly
• 3. con1, tot1 ⊆ B+×B−

Theorem

(L+, L−; CONcon1, TOTtot1) is a d-frame if Closure under con-operations in L+×L− Closure under tot-operations in L+×L− DCPO↑ ↓DL∨,∧ con1 ↑DL∨,∧ tot1

13 / 16

Free d-frame construction

For (B+, ≈+, B−, ≈−, con1, tot1) where

• 1. L+ = Fr B+ /≈+
• 2. L− defined similarly
• 3. con1, tot1 ⊆ B+×B−

Theorem

(L+, L−; CONcon1, TOTtot1) is a d-frame if Closure under con-operations in L+×L− Closure under tot-operations in L+×L− DCPO↑ ↓DL∨,∧ con1 ↑DL∨,∧ tot1

◮ α ∈ con∨, β ∈ tot∧, β+ ≤ α+ =

⇒ α− ≤ β−

◮ (L+×B−) ∩ ↓con∧, ⊆ ↓con∨

(+ symmetric variants)

∀α ∈ con, β ∈ tot: α+ = β+ or = ⇒ α ⊑ β α− = β−

13 / 16

Vietoris functor for d-frames

Md : (L+, L−; con, tot) → (ML+, ML−; CONcon1, TOTtot1) where tot1 = {(a, ♦b), (♦a, b) : (a, b) ∈ tot} con1 = {(a, ♦b), (♦a, b) : (a, b) ∈ con} a ∧ ♦b ≤ ♦(a ∧ b) and ♦0 = 0 because (a, b) ∈ con mimics a ∧ b = 0

14 / 16

Vietoris functor for d-frames

Md : (L+, L−; con, tot) → (ML+, ML−; CONcon1, TOTtot1) where tot1 = {(a, ♦b), (♦a, b) : (a, b) ∈ tot} con1 = {(a, ♦b), (♦a, b) : (a, b) ∈ con} a ∧ ♦b ≤ ♦(a ∧ b) and ♦0 = 0 because (a, b) ∈ con mimics a ∧ b = 0

14 / 16

Facts

• 1. Md is comonadic
• 2. If L is regular, zero-dimensional or compact

regular then also MdL is.

• 3. Points of MdL (i.e. Σd(MdL)), are in bijection

with α ∈ L+×L− such that (A+) (α+ ∨ u+, α−) ∈ tot = ⇒ (u+, α−) ∈ tot (A−) (α+, α− ∨ u−) ∈ tot = ⇒ (α+, u−) ∈ tot

• 4. W ◦ Σd ∼

= Σd ◦ Md

• 5. =

⇒ Coalg(W)op ∼ = Alg(Md)

15 / 16

Conclusion

◮ Free constructions of d-frames. ◮ W and Md are generalisations of V and M for

all our X op ∼ = A

◮ Domain Theory: because biKReg is equivalent

to the category of stably compact spaces, we

• btained an algebraic dual of V for stably

compact spaces (open problem)

16 / 16

Conclusion

◮ Free constructions of d-frames. ◮ W and Md are generalisations of V and M for

all our X op ∼ = A

◮ Domain Theory: because biKReg is equivalent

to the category of stably compact spaces, we

• btained an algebraic dual of V for stably

compact spaces (open problem)

Thank you for your attention!

16 / 16

Axioms of d-frames

◮ con = ↑con and tot = ↓tot ◮ con and tot are (∧, ∨)-subalgebras of L+×Lop

−

◮ con is DCPO-closed ◮ ∀α ∈ con, β ∈ tot:

α+ = β+ or α− = β− = ⇒ α ⊑ β

16 / 16

Topological properties

frames d-frames a∗ : {x|x ∧ a = 0} {x ∈ L−|(a, x) ∈ con} a ⊳ b: b ∨ a∗ = 1 (b, a∗) ∈ tot Regularity: a = {x|x ⊳ a} Zero-dimensionality: a = {x|x ⊳ x ≤ a} Compactness: For all U ⊆ L:

• U = 1 =

⇒ ∃F ⊆fin U s.t.

• F = 1

For all U ⊆ L+×L−:

• U ∈ tot =

⇒ ∃F ⊆fin U s.t.

• F ∈ tot

16 / 16