Duality of upper and lower powerlocales on locally compact locales - PowerPoint PPT Presentation

Duality of upper and lower powerlocales on locally compact locales Tatsuji Kawai University of Padova

Duality of lower and upper powerlocales on locally compact locales Tatsuji Kawai (University of Padova)

Why powerlocales? ◮ Compact and overt ◮ Duality ◮ Basic Picture ◮ Semantics: modal logic, non-determinism

Background Theorem (Hyland 1983) A locale X is locally compact if and only if the exponential S X over Sierpinski locale S exists.

� � Background Theorem (Hyland 1983) A locale X is locally compact if and only if the exponential S X over Sierpinski locale S exists. LKLoc : the category of locally compact locales. Corollary There is an adjunction S ( − ) LKLoc op , LKLoc ⊥ S ( − ) induced by the natural isomorphism LKLoc ( X , S Y ) ∼ = LKLoc ( X × Y , S ) ∼ = LKLoc ( Y × X , S ) ∼ = LKLoc ( Y , S X ) .

� � Background Question What is a monad on LKLoc induced by the adjunction? S ( − ) LKLoc op , LKLoc ⊥ S ( − ) Theorem (Vickers 2004) S S X ∼ = P U P L X ∼ = P L P U X P L : the lower powerlocale monad. P U : the upper powerlocale monad.

� � Background Question What is a monad on LKLoc induced by the adjunction? S ( − ) LKLoc op , LKLoc ⊥ S ( − ) Theorem (Vickers 2004) S S X ∼ = P U P L X ∼ = P L P U X P L : the lower powerlocale monad. P U : the upper powerlocale monad. Proposition (de Brecht & K 2016) P U S X ∼ = S P L X & P L S X ∼ = S P U X .

� � Background Question What is a monad on LKLoc induced by the adjunction? S ( − ) LKLoc op , LKLoc ⊥ S ( − ) Theorem (Vickers 2004) S S X ∼ = P U P L X ∼ = P L P U X P L : the lower powerlocale monad. P U : the upper powerlocale monad. Proposition (de Brecht & K 2016) P U S X ∼ = S P L X & P L S X ∼ = S P U X . So what?

Locales Definition A frame is a poset with arbitrary joins and finite meets that distributes over joins. A frame homomorphism is a function that preserves finite meets and all joins. The category Loc of locales is the opposite of the category of frames. Notations X , Y : locales. Ω X : the frame corresponding to a locale X . Ω Y Ω f − − → Ω X : the frame homomorphism corresponding to a locale map f : X → Y .

Lower and Upper Powerlocales

� � Definition A suplattice is a poset with arbitrary joins. A suplattice homomorphism is a function that preserves all joins. Write SupLat for the category of suplattices. The forgetful functor U : Frm → SupLat has a left adjoint F : SupLat → Frm : ◮ for each suplattice D , there exists a frame F ( D ) and a suplattice homomorphism ι L D : D → F ( D ) , ◮ for any frame Y and a suplattice homomorphism f : D → Y , there exists a unique frame homomorphism f : F ( D ) → Y such that f � Y . F ( D ) ι L D f D

� � � � Lower powerlocales Definition Let X be a locale. The lower powerlocale P L X is the locale corresponding to the frame F ( U (Ω X )) . The lower powerlocales form a monad � P L , η L , µ L � , where η L X and µ L X are given by Ω η L Ω µ L X � Ω X X � Ω P L P L X Ω P L X Ω P L X ι L ι L ι L X PL X X id Ω X ι L � Ω P L X . X Ω X Ω X

� � Definition A preframe is a poset with directed joins and finite meets which distributes over directed joins. A preframe homomorphism is a function that preserves finite meets and directed joins. Write PrFrm for the category of preframes. The forgetful functor U : Frm → PrFrm has a left adjoint H : PrFrm → Frm : ◮ for each preframe D , there exists a frame H ( D ) and a preframe homomorphism ι U D : D → H ( D ) , ◮ for any frame Y and a preframe homomorphism h : D → Y , there exists a unique frame homomorphism h : H ( D ) → Y such that h � Y . H ( D ) ι U D h D

� � � � Upper powerlocales Definition Let X be a locale. The upper powerlocale P U X is the locale corresponding to the frame H ( U (Ω X )) . The upper powerlocales form a monad � P U , η U , µ U � , where η U X and µ U X are given by Ω η U Ω µ U � Ω X X � Ω P U P U X X Ω P U X Ω P U X ι U ι U ι U X PU X X id Ω X ι U � Ω P U X . X Ω X Ω X

Order Enrichment and Distributivity

Order enrichments Definition The category of Poset of posets is a poset enriched category (i.e. homesets are posets), where morphisms are ordered pointwise. Loc is poset-enriched by specialization order given by def f ≤ g ⇐ ⇒ Ω f ≤ Poset Ω g def ⇐ ⇒ ( ∀ y ∈ Y ) Ω f ( y ) ≤ Ω g ( y ) . Definition In a poset enriched category C , a morphism f : X → Y is the left adjoint to g : Y → X , written f ⊣ g , if id X ≤ g ◦ f & f ◦ g ≤ id Y .

� � � � Order enrichments Lemma For any locale X , we have (in Poset ) ◮ ι L X ⊣ Ω η L ⇒ ι L X ◦ Ω η L ( ⇐ X ≤ id Ω P L X ); X ◮ Ω η U X ⊣ ι U ⇒ id Ω P U X ≤ ι U X ◦ Ω η U ( ⇐ X ). X Ω η L Ω η U X � Ω X � Ω X X Ω P L X Ω P U X ι L ι U X X id Ω X id Ω X Ω X Ω X

KZ-monads Definition Let � T , η, µ � be a monad on a poset enriched category C , where T preserves the order on morphisms. Then, T is a KZ -monad (co KZ -monad) if T η X ≤ η TX ( η TX ≤ T η X ). Proposition � P L , η L , µ L � is a KZ-monad and � P L , η U , µ U � is a coKZ-monad.

KZ-monads Definition Let � T , η, µ � be a monad on a poset enriched category C , where T preserves the order on morphisms. Then, T is a KZ -monad (co KZ -monad) if T η X ≤ η TX ( η TX ≤ T η X ). Proposition � P L , η L , µ L � is a KZ-monad and � P L , η U , µ U � is a coKZ-monad. Proposition Let � T , η, µ � be a KZ-monad on a poset enriched category C . Then, the following are equivalent. 1. α : TX → X is a T -algebra; 2. α ⊣ η X & α ◦ η X = id X ; 3. α ◦ η X = id X . In particular, T -algebra structure on X (if it exists) is unique.

� � � � � � � � � � � Distributivity Let � T , η T , µ T � and � S , η S , µ S � be monads. A distributive law of T over S is a natural transformation σ : S ◦ T → T ◦ S which makes the diagrams commutes. S η T η S T S µ T µ S T S ◦ T S ◦ T ◦ T S ◦ T S ◦ S ◦ T S T σ T � S σ σ T ◦ S ◦ T S ◦ T ◦ S σ η T S T η S σ S T σ � µ T S � T ◦ S T µ S T ◦ S T ◦ T ◦ S T ◦ S ◦ S . Then, T ◦ S is a monad with η S η T S η = id − → S − − → T ◦ S , µ T S ◦ S T µ S µ = T ◦ S ◦ T ◦ S T σ S − − → T ◦ T ◦ S ◦ S − − − → T ◦ S ◦ S − − → T ◦ S .

� � � � � � � � � � � Distributivity Let � T , η T , µ T � and � S , η S , µ S � be monads. A distributive law of T over S is a natural transformation σ : S ◦ T → T ◦ S which makes the diagrams commutes. S η T η S T S µ T µ S T S ◦ T S ◦ T ◦ T S ◦ T S ◦ S ◦ T S T σ T � S σ σ T ◦ S ◦ T S ◦ T ◦ S σ η T S T η S σ S T σ � µ T S � T ◦ S T µ S T ◦ S T ◦ T ◦ S T ◦ S ◦ S . Then, T ◦ S is a monad with η S η T S η = id − → S − − → T ◦ S , µ T S ◦ S T µ S µ = T ◦ S ◦ T ◦ S T σ S − − → T ◦ T ◦ S ◦ S − − − → T ◦ S ◦ S − − → T ◦ S . Proposition (Vickers 2004) There is a natural isomorphism P L ◦ P U ∼ = P U ◦ P L which (together with its inverse) is a distributive law of P L over P U and vice versa.

Double powerlocales Definition A double powerlocale P on Loc is the composite P U ◦ P L (equivalently the composite P L ◦ P U ).

Double powerlocales Definition A double powerlocale P on Loc is the composite P U ◦ P L (equivalently the composite P L ◦ P U ). Lemma (Vickers 2004) Every P -algebra is also P L -algebra and P U -algebra. Moreover, P -algebra structure on a object X (if it exists) is unique.

Double powerlocales Definition A double powerlocale P on Loc is the composite P U ◦ P L (equivalently the composite P L ◦ P U ). Lemma (Vickers 2004) Every P -algebra is also P L -algebra and P U -algebra. Moreover, P -algebra structure on a object X (if it exists) is unique. Proof. If P X α − → X is an P -algebra, its P L -algebra structure is P L η U = P X α → P U P L X ∼ X P L X − − − − → X , which is a retract of η L X : X → P L X (note: P L is a KZ -monad). �

Proposition The forgetful functor P - Alg → P L - Alg has a left adjoint:

Proposition The forgetful functor P - Alg → P L - Alg has a left adjoint: ◮ If P L X α − → X is a P L -algebra, P L µ U P U α P P U X ∼ → P L P U X ∼ = P L P U2 X X − − − = P U P L X → P U X − − is a P -algebra and η U X : X → P U X is a P L -algebra morphism;

� � � � Proposition The forgetful functor P - Alg → P L - Alg has a left adjoint: ◮ If P L X α − → X is a P L -algebra, P L µ U P U α P P U X ∼ → P L P U X ∼ = P L P U2 X X − − − = P U P L X → P U X − − is a P -algebra and η U X : X → P U X is a P L -algebra morphism; β ◮ for any P -algebra P Y − → Y and P L -algebra morphism f : X → Y , there is a unique P -algebra morphism f : P U X → Y such that P U f P U X P U Y f η U β L X � Y . X f