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 School of Computer Science Charles University in Prague University of Birmingham AND TACL, 30 June 2017 0 / 16

J´ onsson-Tarski duality Clp For StoneSp op ∼ propositional Bool = logic Ult

J´ onsson-Tarski duality For modal DGF op ∼ MA = logic U op U Clp StoneSp op ∼ Bool = Ult 1 / 16

DGF are V -coalgebras Vietoris functor def ≡ ( K X , V τ ) Let ( X , τ ) be a Stone space. V ( X , τ ) where 1. K X = compact subsets of X 2. V τ is generated by × � V , + ♦ V (for all V ∈ τ ) where � V = { K ∈ K X | K ⊆ V } × ♦ V = { K ∈ K X | 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. def M ( B ) ≡ 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 M is an “algebraic dual” of V Coalg( V ) op ∼ Alg( M ) = U op U Clp StoneSp op V op ∼ Bool M = Ult 4 / 16

New picture M is an “algebraic dual” of V Coalg( V ) op ∼ Alg( M ) = U op U Clp StoneSp op V op ∼ Bool M = Ult 4 / 16

Frames A complete lattice ( L ; � , ∧ , 0 , 1) is a frame iff � � b ∧ ( a i ) = ( b ∧ a i ) i i ( = ⇒ 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 ∧ ( a i ) = ( b ∧ a i ) i i ( = ⇒ complete Heyting algebra) Ω Example Top op Ω( X ; τ ) = ( τ ; � , ∩ , ∅ , X ) Frm ⊥ Σ Ω( f : X → Y ): U �→ f − 1 [ U ] 5 / 16

Frame M Ω KRegSp op ∼ KRegFrm = Σ Compact ♦ a � = ¬ � ( ¬ a ) regular spaces

Frame M Ω KRegSp op ∼ KRegFrm M ? V = Σ Compact ♦ a � = ¬ � ( ¬ a ) regular def spaces M ( L ) ≡ F r � � 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 ↑ � a i ≈ � ( ↑ a i ) ↑ ♦ a i ≈ ♦ ( ↑ a i ) � � � � 6 / 16

The whole perspective StoneSp Priestley PriesSp spaces KRegSp biKReg Compact regular bitopological spaces

The whole perspective ∼ = StoneSp op Bool ∼ Priestley = PriesSp op DLat spaces ∼ = KRegSp op KRegFrm ∼ = biKReg op d - KReg Compact regular Compact regular d-frames bitopological spaces 7 / 16

The whole perspective ∼ = StoneSp op Bool ∼ Priestley = PriesSp op DLat spaces ∼ = KRegSp op KRegFrm ∼ = biKReg op d - KReg ∀X ∈ { StoneSp , KRegSp , PriesSp } ∃ V : X → X Compact regular Compact regular d-frames bitopological spaces ∀A ∈ { Bool , KRegFrm , DLat } ∃ M : A → A Coalg( V ) op ∼ (whenever X op ∼ and = Alg( M ) = A ) 7 / 16

The task: generalise V ’s and M ’s V op X op ∼ A M = I op J biKReg op W op ? ∼ d - KReg M d ? = such that I ◦ V ∼ = W ◦ I and J ◦ M ∼ = M d ◦ 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 − ( x + , x − ) ∈ con , x ′ + ≤ x + , x ′ − ≤ x − (+ axioms, e.g. ) ( x ′ + , x ′ − ) ∈ con Example Ω d ( X , τ + , τ − ) = ( τ + , τ − , con X , tot X ) def ( U , V ) ∈ con X ≡ U ∩ V = ∅ def ( U , V ) ∈ tot X ≡ 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 − ( x + , x − ) ∈ con , x ′ + ≤ x + , x ′ − ≤ x − (+ axioms, e.g. ) ( x ′ + , x ′ − ) ∈ con Ω d biTop op ⊥ d - Frm Σ d 9 / 16

Example: embedding the frame duality KRegSp op ∼ KRegFrm = I op J biKReg op ∼ d - KReg = ◮ I : ( X , τ ) �→ ( X , τ, τ ) ◮ J : L �→ ( L , L , con L , tot L ) where ( a , b ) ∈ con L iff a ∧ b = 0 ( a , b ) ∈ tot L iff a ∨ b = 1 10 / 16

Example: embedding the Priestley duality PriesSp op ∼ DLat = I op J biKReg op ∼ d - KReg = ◮ I : ( X , τ, ≤ ) �→ ( X , τ + , τ − ) where τ + = Up( X , ≤ ) ∩ τ τ − = Down( X , ≤ ) ∩ τ ◮ J : D �→ (Idl( D ) , Filt( D ) , con D , tot D ) where ( I , F ) ∈ con D ∀ i ∈ I , f ∈ F : i ≤ f iff ( I , F ) ∈ tot D I ∩ F � = ∅ iff 11 / 16

Ω d biKReg op ∼ d - KReg M d ? W ? = Σ d 12 / 16

Ω d biKReg op ∼ d - KReg M d ? W ? = Σ d W : ( X ; τ + , τ − ) �→ ( K c X ; V τ + , V τ − ) where 1. K c X = compact convex subsets of X (Note: ( ≤ τ + ) = ( ≥ τ − )) 2. V τ + is generated by × � U + , + ♦ U + (for all U + ∈ τ + ) 3. V τ − is generated by × � U − , + ♦ U − (for all U − ∈ τ − ) 12 / 16

Free d-frame construction For ( B + , ≈ + , B − , ≈ − , con 1 , tot 1 ) where 1. L + = F r � B + � / ≈ + 2. L − defined similarly 3. con 1 , tot 1 ⊆ B + × B − Theorem ( L + , L − ; CON � con 1 � , TOT � tot 1 � ) is a d-frame if Closure under ↑ DL ∨ , ∧ � tot 1 � Closure under con -operations tot -operations in L + × L − in L + × L − DCPO � ↑ �↓ DL ∨ , ∧ � con 1 �� 13 / 16

Free d-frame construction For ( B + , ≈ + , B − , ≈ − , con 1 , tot 1 ) where 1. L + = F r � B + � / ≈ + 2. L − defined similarly 3. con 1 , tot 1 ⊆ B + × B − Theorem ( L + , L − ; CON � con 1 � , TOT � tot 1 � ) is a d-frame if Closure under ↑ DL ∨ , ∧ � tot 1 � Closure under con -operations tot -operations in L + × L − in L + × L − DCPO � ↑ �↓ DL ∨ , ∧ � con 1 �� 13 / 16

Free d-frame construction For ( B + , ≈ + , B − , ≈ − , con 1 , tot 1 ) where 1. L + = F r � B + � / ≈ + 2. L − defined similarly ∀ α ∈ con , β ∈ tot: 3. con 1 , tot 1 ⊆ B + × B − α + = β + or ⇒ α ⊑ β = α − = β − Theorem ( L + , L − ; CON � con 1 � , TOT � tot 1 � ) is a d-frame if ◮ α ∈ con ∨ , β ∈ tot ∧ , β + ≤ α + = ⇒ α − ≤ β − ◮ ( L + × B − ) ∩ ↓ con ∧ , � ⊆ ↓ con ∨ Closure under ↑ DL ∨ , ∧ � tot 1 � Closure under (+ symmetric variants) con -operations tot -operations in L + × L − in L + × L − DCPO � ↑ �↓ DL ∨ , ∧ � con 1 �� 13 / 16

Vietoris functor for d-frames M d : ( L + , L − ; con , tot) �→ ( M L + , M L − ; CON � con 1 � , TOT � tot 1 � ) where tot 1 = { ( � a , ♦ b ) , ( ♦ a , � b ) : ( a , b ) ∈ tot } con 1 = { ( � 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 M d : ( L + , L − ; con , tot) �→ ( M L + , M L − ; CON � con 1 � , TOT � tot 1 � ) where tot 1 = { ( � a , ♦ b ) , ( ♦ a , � b ) : ( a , b ) ∈ tot } con 1 = { ( � 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. M d is comonadic 2. If L is regular, zero-dimensional or compact regular then also M d L is. 3. Points of M d L (i.e. Σ d ( M d L )), 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 ◦ Σ d ∼ = Σ d ◦ M d ⇒ Coalg( W ) op ∼ 5. = = Alg( M d ) 15 / 16

Conclusion ◮ Free constructions of d-frames. ◮ W and M d 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 obtained an algebraic dual of V for stably compact spaces (open problem) 16 / 16

Conclusion ◮ Free constructions of d-frames. ◮ W and M d 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 obtained 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 + × L op − ◮ 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 } b ∨ a ∗ = 1 ( b , a ∗ ) ∈ tot a ⊳ b : 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