Prelude Duality A Final V -coalgebra Future Work A final Vietoris coalgebra beyond compact spaces and a generalized J´ onsson-Tarski duality Liang-Ting Chen School of Computer Science, University of Birmingham 26 July 2011
� � � � � Prelude Duality A Final V -coalgebra Future Work J´ onsson-Tarski duality is . . . 1 in old days . . . MA ( BA ) DGF U U Ultra � Stone BA Clop
� � � � � � � � � � Prelude Duality A Final V -coalgebra Future Work J´ onsson-Tarski duality is . . . 1 in old days . . . MA ( BA ) DGF U U Ultra � Stone BA Clop 2 nowadays . . . Alg ( M ) Coalg ( V ) U U Pt Idl � Stone BA K Ω
� � � � � Prelude Duality A Final V -coalgebra Future Work Is is true in general? The main purpose of this talk: Alg ( M ) Coalg ( V ) Pt � Top Frm Ω and an application, the final V -coalgebra. Let’s see how far we can go.
Prelude Duality A Final V -coalgebra Future Work Kripke frames are P -coalgebras Definition A Kripke frame � X , R � consists of 1 a set X and 2 a relation R of X ξ R : X → P X x �→ { y : xRy }
� � � Prelude Duality A Final V -coalgebra Future Work Bounded morphisms are P -coalgebra morphisms Definition A bounded morphism f : � X , R � → � Y , S � is a functional bisimulation. ξ X P X f P f η � P Y Y That is, f is a P -coalgebra morphism.
Prelude Duality A Final V -coalgebra Future Work Descriptive general frames are V -coalgebras Definition A descriptive general frame is 1 a Kripke frame � X , ξ : X → P X � 2 on a Stone space � X , B�
Prelude Duality A Final V -coalgebra Future Work Descriptive general frames are V -coalgebras Definition A descriptive general frame is 1 a Kripke frame � X , ξ : X → P X � 2 on a Stone space � X , B� 3 ξ ( x ) ∈ K X 4 B is closed under � V = { x : ξ ( x ) ⊆ V } 1 ♦ V = { x : ξ ( x ) ∩ V � = ∅} . 2
Prelude Duality A Final V -coalgebra Future Work Descriptive general frames are V -coalgebras Definition A descriptive general frame is 1 a Kripke frame � X , ξ : X → P X � 2 on a Stone space � X , B� 3 ξ ( x ) ∈ K X 4 B is closed under � V = { x : ξ ( x ) ⊆ V } 1 ♦ V = { x : ξ ( x ) ∩ V � = ∅} . 2 ξ i : � X , B i � → �K X , ? � What is the finest topology making every ξ i continuous?
Prelude Duality A Final V -coalgebra Future Work Definition A Vietoris topology V X = �K X , τ � of a Stone space X is a Stone space with τ generated by 1 � V = { K ∈ K X : K ⊆ V } 2 ♦ V = { K ∈ K X : K ∩ V � = ∅} where V ∈ K Ω X .
Prelude Duality A Final V -coalgebra Future Work Definition A Vietoris topology V X = �K X , τ � of a Stone space X is a Stone space with τ generated by 1 � V = { K ∈ K X : K ⊆ V } 2 ♦ V = { K ∈ K X : K ∩ V � = ∅} where V ∈ K Ω X . ξ − 1 � Ω X � � Ω V X K Ω X ξ − 1 ( � V ) = { x : ξ ( x ) ∈ � V } = { x : ξ ( x ) ⊆ V } similarly for ξ − 1 ( ♦ V ).
Prelude Duality A Final V -coalgebra Future Work Bounded morphisms are V -coalgebra morphisms Definition A morphism f : � X , R , B� → � Y , S , C� is a bounded morphism and also f − 1 ( C ) ∈ B for any C ∈ C (continuity). Fact V f ( K ) = f [ K ] is compact V f is continuous.
� � � � � � Prelude Duality A Final V -coalgebra Future Work Bounded morphisms are V -coalgebra morphisms Definition A morphism f : � X , R , B� → � Y , S , C� is a bounded morphism and also f − 1 ( C ) ∈ B for any C ∈ C (continuity). Fact V f ( K ) = f [ K ] is compact V f is continuous. ξ ξ X P X X V X f P f f V f η � P Y � V Y Y Y η
Prelude Duality A Final V -coalgebra Future Work Modal algebras are M -algebras Definition Modal algebra = Boolean algebra + unary operators � , ♦ subject to normal modal logic laws
Prelude Duality A Final V -coalgebra Future Work Modal algebras are M -algebras Definition Modal algebra = Boolean algebra + unary operators � , ♦ subject to normal modal logic laws Definition A modal algebra construction M A of a Boolean algebra is BA � ♦ A ∪ � A | � ( a ∧ b ) = � a ∧ � b ♦ ( a ∨ b ) = ♦ a ∨ ♦ b ♦ ( a ∧ b ) ≥ ♦ a ∧ � b � ( a ∨ b ) ≤ ♦ a ∨ � b �
Prelude Duality A Final V -coalgebra Future Work A M -algebra is an interpretation: � A �→ � A , � , ♦ � M A α � a = α ( � a ) , ♦ a = α ( ♦ a ).
� � Prelude Duality A Final V -coalgebra Future Work A M -algebra is an interpretation: � A �→ � A , � , ♦ � M A α � a = α ( � a ) , ♦ a = α ( ♦ a ). Conversely, we can obtain an interpretation by freeness: � id � A M A � � � � � a �→ � a , ♦ a �→ ♦ a � � � id � � � A , � , ♦ �
� � Prelude Duality A Final V -coalgebra Future Work Modal algebra morphisms are M -algebra morphisms Definition A modal algebra morphism f is a Boolean algebra morphism and f ( � A a ) = � B f ( a ), f ( ♦ A a ) = ♦ B f ( a ) and relations. � A � A A f f � B B � B Modal algebra morphisms are M -algebras morphisms.
� � � � Prelude Duality A Final V -coalgebra Future Work Definition Given f : A → B , define 1 M f ( � a ) = � f ( a ) and M f ( ♦ a ) = ♦ f ( a ) 2 M f is a Boolean algebra homomorphism by freeness of M A . α M A A M f f � B M B β f ( � A a ) = ( f ◦ α )( � a ) = β ( � f ( a )) = � B f ( a )
Prelude Duality A Final V -coalgebra Future Work Short summary Fact 1 DGF ∼ = Coalg ( V ) and MA ( BA ) ∼ = Alg ( M ) onsson-Tarski duality: DGF ∼ 2 The classical J´ = MA ( BA ) op 3 The (co)-algebra viewpoint: Coalg ( V ) ∼ = Alg ( M ) op
� � � � � � Prelude Duality A Final V -coalgebra Future Work Short summary Fact 1 DGF ∼ = Coalg ( V ) and MA ( BA ) ∼ = Alg ( M ) onsson-Tarski duality: DGF ∼ 2 The classical J´ = MA ( BA ) op 3 The (co)-algebra viewpoint: Coalg ( V ) ∼ = Alg ( M ) op Question 1 What is the relationship between M and V ? 2 An extension of M and V ? M V BA BA Stone Stone Idl Idl J J � Frm � Top Frm Top M F V ′
Prelude Duality A Final V -coalgebra Future Work Duality between algebras and coalgebras Fact Coalg ( T ) op ≡ Alg ( T op ) where T op : X op → X op x �→ x f op �→ ( Tf ) op
� � � � � � � Prelude Duality A Final V -coalgebra Future Work Dual functors T is dual to L if F � X op A T op L X op A F i.e. LF ∼ = FT . Fact Coalg ( T ) op ∼ = Alg ( T op ) ∼ = Alg ( L ) if X op ∼ = A and LF ∼ = FT.
� Prelude Duality A Final V -coalgebra Future Work Topological spaces and Frames A dual adjunction . . . Pt � Top Frm Ω Ω X = the complete lattice of open sets with � � S ∧ a = ( s ∧ a ) s ∈ S
� Prelude Duality A Final V -coalgebra Future Work Topological spaces and Frames A dual adjunction . . . Pt � Top Frm Ω Ω X = the complete lattice of open sets with � � S ∧ a = ( s ∧ a ) s ∈ S Pt A = the space of frame homomorphisms f : A → 2 with open sets U a = { ϕ ∈ Pt A : ϕ ( a ) = ⊤} or, neighbourhoods systems.
Prelude Duality A Final V -coalgebra Future Work Sober spaces and spatial frames Definition A frame A is spatial if the unit is an iso. A space X is sober if the unit is an iso.
� Prelude Duality A Final V -coalgebra Future Work Sober spaces and spatial frames Definition A frame A is spatial if the unit is an iso. A space X is sober if the unit is an iso. A cheat dual equivalence . . . Pt � Sob SFrm Ω
Prelude Duality A Final V -coalgebra Future Work Definition A modal algebra construction M F A of a frame A is Frm � � A ∪ ♦ A | normal modal logic laws with the following � for any directed S ⊆ A 1 � ( � S ) = � s ∈ S � s 2 ♦ ( � S ) = � s ∈ S ♦ s
� � � Prelude Duality A Final V -coalgebra Future Work Definition A modal algebra construction M F A of a frame A is Frm � � A ∪ ♦ A | normal modal logic laws with the following � for any directed S ⊆ A 1 � ( � S ) = � s ∈ S � s 2 ♦ ( � S ) = � s ∈ S ♦ s M BA BA Idl Idl � Frm Frm M F M F is an extension of M along Idl .
Prelude Duality A Final V -coalgebra Future Work Stably locally compact frames Unfortunately, SFrm is not closed under M F . Stably locally compact frames (locales) are closed under M F , i.e. M F : SLCFrm → SLCFrm . Definition A stably locally compact frame is 1 a continuous domain 2 x ≪ y 1 and x ≪ y 2 ⇒ x ≪ y 1 ∧ y 2
Prelude Duality A Final V -coalgebra Future Work Stably locally compact frames Unfortunately, SFrm is not closed under M F . Stably locally compact frames (locales) are closed under M F , i.e. M F : SLCFrm → SLCFrm . Definition A stably locally compact frame is 1 a continuous domain 2 x ≪ y 1 and x ≪ y 2 ⇒ x ≪ y 1 ∧ y 2 Example 1 the ideal completion of Boolean algebras 2 the ideal completion of distributive lattices 3 compact regular frames 4 . . .
Prelude Duality A Final V -coalgebra Future Work Stably locally compact spaces By dual equivalence, SLCFrm ∼ = SLCSp op : Definition A space X is stably locally compact if X ∈ Sob and Ω X ∈ SLCFrm .
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