Stone Spaces versus Priestley Spaces

Margarida Dias

Mathematics Department, University of the Azores Centre of Mathematics of University of Coimbra

Coimbra, October 26-28,2007

Margarida Dias (Univ. Azores) 1 / 20 Priestley (PSp)/Stone as categories which arise from an equivalence induced by a ”bigger”adjunction between OrdT op and Lat/T op and Lat;

PSp = Profinite preorder + order/ Stone = Profinite;

PSp=OrdComp2do /Stone=Comp2d .

Stone Spaces versus Priestley Spaces

We describe three situations where the Stone spaces naturally replace the Priestley spaces:

Margarida Dias (Univ. Azores) 2 / 20 PSp = Profinite preorder + order/ Stone = Profinite;

PSp=OrdComp2do /Stone=Comp2d .

Stone Spaces versus Priestley Spaces

We describe three situations where the Stone spaces naturally replace the Priestley spaces:

Priestley (PSp)/Stone as categories which arise from an equivalence induced by a ”bigger”adjunction between OrdT op and Lat/T op and Lat;

Margarida Dias (Univ. Azores) 2 / 20 PSp=OrdComp2do /Stone=Comp2d .

Stone Spaces versus Priestley Spaces

We describe three situations where the Stone spaces naturally replace the Priestley spaces:

Priestley (PSp)/Stone as categories which arise from an equivalence induced by a ”bigger”adjunction between OrdT op and Lat/T op and Lat;

PSp = Profinite preorder + order/ Stone = Profinite;

Margarida Dias (Univ. Azores) 2 / 20 Stone Spaces versus Priestley Spaces

We describe three situations where the Stone spaces naturally replace the Priestley spaces:

Priestley (PSp)/Stone as categories which arise from an equivalence induced by a ”bigger”adjunction between OrdT op and Lat/T op and Lat;

PSp = Profinite preorder + order/ Stone = Profinite;

PSp=OrdComp2do /Stone=Comp2d .

Margarida Dias (Univ. Azores) 2 / 20 Priestley Duality

Recall that Every adjunction F a U : A → B(η, ), induces a largest equivalence between the full subcategories A0 of A and B0 of B where

A0 = Fixε ≡ {A ∈ A|εAis an }

B0 = Fixη ≡ {B ∈ B|ηB is an isomorphism}

Margarida Dias (Univ. Azores) 3 / 20 Identifying: −1 f ∈ Hom(L, 2l ) with f (1) (prime filter);

F (L) = (Fp(L), τ, ⊆), L ∈ Lat

S = {Ub|b ∈ L} ∪ {Fp(L) − Ub|b ∈ L} with Ub = {F ∈ Fp(L)|b ∈ F }

Priestley Duality

2l = {0 < 1} 2do = ({0 < 1}, D)

Margarida Dias (Univ. Azores) 4 / 20 Priestley Duality

2l = {0 < 1} 2do = ({0 < 1}, D)

Identifying: −1 f ∈ Hom(L, 2l ) with f (1) (prime filter);

F (L) = (Fp(L), τ, ⊆), L ∈ Lat

S = {Ub|b ∈ L} ∪ {Fp(L) − Ub|b ∈ L} with Ub = {F ∈ Fp(L)|b ∈ F }

Margarida Dias (Univ. Azores) 4 / 20 Priestley Duality

2l = {0 < 1} 2do = ({0 < 1}, D)

Identifying: −1 g ∈ Hom(X , 2do) with g (0) (decreasing clopen); U(X ) = (DClopen(X )∩, ∪), X ∈ OrdT op

Margarida Dias (Univ. Azores) 5 / 20 Priestley Duality

The F is left adjoint to U : OrdT opop → Lat

ηL : L → UF (L), ηL(a) = Γa = {F ∈ Fp(L)|a ∈ F }

X (x) = Σx = {A ∈ DClopen(X )|x ∈ A}

Margarida Dias (Univ. Azores) 6 / 20 (Fp(L), τ, ⊆) is a Priestley space for every L.

If X is a Priestley space then X is an isomorphism. (DClopen(X ), ∩, ∪) is a distributive lattice for every space X .

If L is a distributive lattice then ηL is an isomorphism. We obtain the Priestley Duality as the equivalence induced by the adjunction above.

Priestley Duality

By well-known results, namely:

Margarida Dias (Univ. Azores) 7 / 20 (DClopen(X ), ∩, ∪) is a distributive lattice for every space X .

If L is a distributive lattice then ηL is an isomorphism. We obtain the Priestley Duality as the equivalence induced by the adjunction above.

Priestley Duality

By well-known results, namely:

(Fp(L), τ, ⊆) is a Priestley space for every distributive lattice L.

If X is a Priestley space then X is an isomorphism.

Margarida Dias (Univ. Azores) 7 / 20 Priestley Duality

By well-known results, namely:

(Fp(L), τ, ⊆) is a Priestley space for every distributive lattice L.

If X is a Priestley space then X is an isomorphism. (DClopen(X ), ∩, ∪) is a distributive lattice for every space X .

If L is a distributive lattice then ηL is an isomorphism. We obtain the Priestley Duality as the equivalence induced by the adjunction above.

Margarida Dias (Univ. Azores) 7 / 20 Priestley Duality

Fix = PSp Fixη = DLat

Margarida Dias (Univ. Azores) 8 / 20 U(X ) = (Clopen(X ), ∩, ∪), X ∈ T op, is a Boolean algebra.

F (L) = (Fp(L), τ), L ∈ Lat, is a .

Stone Duality

In a similar way, from the facts

2l = {0 < 1}

2d = ({0, 1}, D)

Margarida Dias (Univ. Azores) 9 / 20

In a similar way, from the facts

2l = {0 < 1}

2d = ({0, 1}, D)

U(X ) = (Clopen(X ), ∩, ∪), X ∈ T op, is a Boolean algebra.

F (L) = (Fp(L), τ), L ∈ Lat, is a Stone space.

Margarida Dias (Univ. Azores) 9 / 20 Stone Duality

Fix = Stone Fixη = Bool

Margarida Dias (Univ. Azores) 10 / 20 Profinite spaces are exactly the Stone spaces.

We are going to show that the Priestley spaces are limits of finite topologically-discrete preordered spaces.

Profinite Preorders / Profinite Spaces

Profinite orders are the Priestley spaces.

Margarida Dias (Univ. Azores) 11 / 20 We are going to show that the Priestley spaces are limits of finite topologically-discrete preordered spaces.

Profinite Preorders / Profinite Spaces

Profinite orders are the Priestley spaces.

Profinite spaces are exactly the Stone spaces.

Margarida Dias (Univ. Azores) 11 / 20 Profinite Preorders / Profinite Spaces

Profinite orders are the Priestley spaces.

Profinite spaces are exactly the Stone spaces.

We are going to show that the Priestley spaces are limits of finite topologically-discrete preordered spaces.

Margarida Dias (Univ. Azores) 11 / 20 The unique morphism ϕ, for every R ∈ R, is an homeomorphism.

Profinite Preorders / Profinite Spaces

X ∈ Stone

R - Set of all equivalence relations R of X such that X /R is finite and topologically-discrete.

D : R → Stone, D(R) = X /R

Margarida Dias (Univ. Azores) 12 / 20 Profinite Preorders / Profinite Spaces

X ∈ Stone

R - Set of all equivalence relations R of X such that X /R is finite and topologically-discrete.

D : R → Stone, D(R) = X /R

The unique morphism ϕ, for every R ∈ R, is an homeomorphism.

Margarida Dias (Univ. Azores) 12 / 20 The full subcategory of PreordT op:

PreordP - Stone spaces equipped with a preorder with respect to which they are totally preordered-disconnected.

Profinite Preorders / Profinite Spaces

We remark that:

X ∈ PSp

The relation induced in X /R by transitive closure of the image of the relation on X is not, in general, an order relation.

Margarida Dias (Univ. Azores) 13 / 20 Profinite Preorders / Profinite Spaces

We remark that:

X ∈ PSp

The relation induced in X /R by transitive closure of the image of the relation on X is not, in general, an order relation.

The full subcategory of PreordT op:

PreordP - Stone spaces equipped with a preorder with respect to which they are totally preordered-disconnected.

Margarida Dias (Univ. Azores) 13 / 20 We just have to prove that the morphism ϕ is an order isomorphism, for every R ∈ R.

Profinite Preorders / Profinite Spaces

X ∈ PreordP

R - Set of all equivalence relations R of X such that X /R is finite, topologically-discrete and equipped with the preorder induced by image of the preorder of X .

DO : R → PreordP, DO (R) = X /R

Margarida Dias (Univ. Azores) 14 / 20 Profinite Preorders / Profinite Spaces

X ∈ PreordP

R - Set of all equivalence relations R of X such that X /R is finite, topologically-discrete and equipped with the preorder induced by image of the preorder of X .

DO : R → PreordP, DO (R) = X /R

We just have to prove that the morphism ϕ is an order isomorphism, for every R ∈ R.

Margarida Dias (Univ. Azores) 14 / 20 PreordP is the of profinite preorders.

Profinite Preorders / Profinite Spaces

For that it is enough to show that if x 6 x0 U is the clopen decreasing subset of X such that x0 ∈ U and x ∈/ U

RU the equivalence relation on X corresponding to the partition

X = U ∪ (X − U)

0 0 then pRU separates x and x and so ϕ(x) 6 ϕ(x )

Margarida Dias (Univ. Azores) 15 / 20 Profinite Preorders / Profinite Spaces

For that it is enough to show that if x 6 x0 U is the clopen decreasing subset of X such that x0 ∈ U and x ∈/ U

RU the equivalence relation on X corresponding to the partition

X = U ∪ (X − U)

0 0 then pRU separates x and x and so ϕ(x) 6 ϕ(x )

PreordP is the category of profinite preorders.

Margarida Dias (Univ. Azores) 15 / 20 X ∈ PreordT op

X is a Priestley space if and only if the limit object of DO : R → PreordP is an ordered space.

Profinite Preorders / Profinite Spaces

PSp, being a full regular-epireflective subcategory of PreordP, is closed under limits.

Margarida Dias (Univ. Azores) 16 / 20 PSp, being a full regular-epireflective subcategory of PreordP, is closed under limits.

Profinite Preorders / Profinite Spaces

X ∈ PreordT op

X is a Priestley space if and only if the limit object of DO : R → PreordP is an ordered space.

Margarida Dias (Univ. Azores) 16 / 20 Using the terminology of Engelking and Mr´oka:

The E - completely regular space are the subspaces of some power of E. The E - are the closed subspaces of some power of E.

Full subcategories of T op:

CRegE : E - completely regular spaces. CompE : E - compact spaces.

Stone Spaces are the 2d - compact spaces

Let E be an Hausdorff space.

Margarida Dias (Univ. Azores) 17 / 20 Full subcategories of T op:

CRegE : E - completely regular spaces. CompE : E - compact spaces.

Stone Spaces are the 2d - compact spaces

Let E be an Hausdorff space. Using the terminology of Engelking and Mr´oka:

The E - completely regular space are the subspaces of some power of E. The E - compact space are the closed subspaces of some power of E.

Margarida Dias (Univ. Azores) 17 / 20 Stone Spaces are the 2d - compact spaces

Let E be an Hausdorff space. Using the terminology of Engelking and Mr´oka:

The E - completely regular space are the subspaces of some power of E. The E - compact space are the closed subspaces of some power of E.

Full subcategories of T op:

CRegE : E - completely regular spaces. CompE : E - compact spaces.

Margarida Dias (Univ. Azores) 17 / 20 CReg2d - category of Hausdorff zero-dimensional spaces,

Comp2d - category of Stone spaces,

as proved by B. Banaschewski.

Stone Spaces are the 2- compact spaces

E = 2d ( The two point discrete )

Margarida Dias (Univ. Azores) 18 / 20 Stone Spaces are the 2- compact spaces

E = 2d ( The two point discrete topological space)

CReg2d - category of Hausdorff zero-dimensional spaces,

Comp2d - category of Stone spaces,

as proved by B. Banaschewski.

Margarida Dias (Univ. Azores) 18 / 20 We consider the full subcategories of OrdT op:

OrdCRegE : E - completely regular spaces with the induced order. OrdCompE : E - compact spaces with the induced order.

OrdCompE is a reflective subcategory of OrdCRegE .

Priestley Spaces are the 2do - ordered compact spaces

Let E be an ordered Hausdorff space.

Margarida Dias (Univ. Azores) 19 / 20 OrdCompE is a reflective subcategory of OrdCRegE .

Priestley Spaces are the 2do - ordered compact spaces

Let E be an ordered Hausdorff space.

We consider the full subcategories of OrdT op:

OrdCRegE : E - completely regular spaces with the induced order. OrdCompE : E - compact spaces with the induced order.

Margarida Dias (Univ. Azores) 19 / 20 Priestley Spaces are the 2do - ordered compact spaces

Let E be an ordered Hausdorff space.

We consider the full subcategories of OrdT op:

OrdCRegE : E - completely regular spaces with the induced order. OrdCompE : E - compact spaces with the induced order.

OrdCompE is a reflective subcategory of OrdCRegE .

Margarida Dias (Univ. Azores) 19 / 20 OrdComp2do is the category PSp

Priestley Spaces are the 2do- ordered compact spaces

E = 2do (The two chain with discrete topology )

Margarida Dias (Univ. Azores) 20 / 20 Priestley Spaces are the 2do- ordered compact spaces

E = 2do (The two chain with discrete topology )

OrdComp2do is the category PSp

Margarida Dias (Univ. Azores) 20 / 20