extending stone-priestley duality along full embeddings

Extending Stone-Priestley duality

along full embeddings

Celia Borlidobased on joint work with A. L. Suarez

Centre for Mathematics, University of Coimbra

BLAST 2021

9 - 13 June

1. The spatial-sober duality

(and its restriction to duality for bounded distributive lattices)

2. The categories of Pervin spaces and of Frith frames

3. Extending Stone-Priestley duality along full embeddings

4. The bitopological point of view

Top Frmop

Priestley DLatop

notfull

Pervin Frithop

CPervin CFrithop

∼= ∼=

Top Frmop

Priestley DLatop

notfull

Pervin Frithop

CPervin CFrithop

∼= ∼=

Top Frmop

Priestley DLatop

notfull

Pervin Frithop

CPervin CFrithop

∼= ∼=

Top Frmop

Priestley DLatop

notfull

Pervin Frithop

CPervin CFrithop

∼= ∼=

Top Frmop

Priestley DLatop

notfull

Pervin Frithop

CPervin CFrithop

∼= ∼=

The spatial-sober duality

Frames

A frame is a complete lattice L satisfying

a ∧∨i∈I

bi =∨

(a ∧ bi ).

Frame homomorphisms preserve

finite meets and arbitrary joins.

Topological spaces

and continuous functions.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 2

I Ω : Top→ Frmop

Ω(X ) := (open subsets of X,⊆) Ω(Xf−→ Y ) := (Ω(Y )

f −1

−−→ Ω(X ))

I Σ : Frmop → Top

Σ(L) := c. p. filters of L

a := F | a ∈ F, a ∈ L

Σ(Lh−→ M) := (Σ(M)

−−→ Σ(L))

We have an adjunction Ω : Top Frmop : Σ which restricts and

co-restricts to a duality between sober spaces and spatial frames.

− Spatial frame: a frame of the form Ω(X ) for some topological space X .

− Sober space: a space that is completely determined by its set of open subsets.

I Ω : Top→ Frmop

f −1

−−→ Ω(X ))

a := F | a ∈ F, a ∈ L

Σ(Lh−→ M) := (Σ(M)

−−→ Σ(L))

I Ω : Top→ Frmop

f −1

−−→ Ω(X ))

a := F | a ∈ F, a ∈ L

Σ(Lh−→ M) := (Σ(M)

−−→ Σ(L))

Bounded distributive lattices seen as coherent frames

A coherent frame is a frame L whose set of compact elements K (L) is

closed under finite meets (thus, a sublattice) and join-dense in L.

Coherent homomorphisms are frame homomorphisms that preserve

compact elements.

I If D is a bounded distributive lattice, (Idl(D),⊆) is a coherent frame.

If h : C → D is a lattice homomorphism, Idl(h) : (J ∈ Idl(C )) 7→ 〈h[J]〉Idlis a coherent homomorphism.

I If L is a coherent frame, K (L) is a bounded distributive lattice.

If h : L→ M is a coherent homomorphism, the restriction and

co-restriction K (h) : K (L)→ K (M) of h is a lattice homomorphism.

These assignments define an equivalence of categories DLat ∼= CohFrm

compact elements.

Spectral spaces

A spectral space is a T0 compact sober space (X , τ) whose set of

compact open subsets K o(X , τ) is closed under finite meets and is a

basis for the topology.

Spectral maps are continuous functions such that the preimage of

compact open subsets is again compact (and open).

The adjunction Ω : Top Frmop : Σ restricts and co-restricts to an

equivalence

Spec ∼= CohFrmop

Spectral spaces

equivalence

Spec ∼= CohFrmop

Spectral spaces

equivalence

Spec ∼= CohFrmop

Stone-Priestley duality for bounded distributive lattices

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

Priestley

Pervin Frithop

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

Pervin spaces

What are the quasi-uniformizable topological spaces?

Every topology comes from a quasi-uniformity!

Every topology comes from a transitive and totally bounded quasi-uniformity.

Pervin spaces

A Pervin space is a pair (X ,S), where X is a set and S ⊆ P(X ) is a

bounded sublattice.

A morphism of Pervin spaces f : (X ,S)→ (Y , T ) is a map f : X → Y

such that for every T ∈ T , we have f −1(T ) ∈ S.

There is a full embedding Top → Pervin, (X , τ) 7→ (X , τ).

Theorem (Pin, 2017)

The category of transitive and totally bounded quasi-uniform spaces is

equivalent to the category Pervin of Pervin spaces.

Pervin spaces

bounded sublattice.

Theorem (Pin, 2017)

Pervin spaces

bounded sublattice.

Theorem (Pin, 2017)

Frith frames

A Frith frame is a pair (L, S), where L is a frame and S ⊆ L is a

join-dense bounded sublattice.

A morphism of Frith frames h : (L,S)→ (M,T ) is a frame

homomorphism h : L→ S such that h[S ] ⊆ T .

There is a full embedding Frm → Frith, L 7→ (L, L).

Theorem (B., Suarez)

The category Frith of Frith frames is a full subcategory of the category

of transitive and totally bounded quasi-uniform frames.

Frith frames

A morphism of Frith frames h : (L, S)→ (M,T ) is a frame

Frith frames

A morphism of Frith frames h : (L, S)→ (M,T ) is a frame

The Frith-Pervin adjunction

I Ω : Pervin→ Frithop

Ω(X ,S) := (〈S〉Frm, S)

Ω((X ,S)f−→ (Y , T )) := (Ω(Y , T )

f −1

−−→ Ω(X ,S))

I Σ : Frithop → Pervin

Σ(L,S) := (Σ(L), s | s ∈ S) (s := F | s ∈ F)

Σ((L,S)h−→ (M,T )) := (Σ(M,T )

−−→ Σ(L,S))

Pervin Frithop

Top Frmop

The Frith-Pervin adjunction

I Ω : Pervin→ Frithop

Ω(X ,S) := (〈S〉Frm, S)

Ω((X ,S)f−→ (Y , T )) := (Ω(Y , T )

f −1

−−→ Ω(X ,S))

I Σ : Frithop → Pervin

Σ(L,S) := (Σ(L), s | s ∈ S) (s := F | s ∈ F)

Σ((L,S)h−→ (M,T )) := (Σ(M,T )

−−→ Σ(L,S))

Pervin Frithop

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

CPervin CFrithop

≡ ≡

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

CPervin CFrithop

≡ ≡

Symmetric Frith frames

A Frith frame (L,S) is symmetric if S is a Boolean algebra.

Proposition (B., Suarez)

Symmetric Frith frames form a full reflective subcategory of Frith.

That is, for every Frith frame (L,S), there exists a symmetric Frith frame

Sym(L,S) = (C(L,S), 〈S〉BA),

called the symmetrization of (L,S), such that for every h : (L,S)→ (M,B) with

(M,B) symmetric there is a unique h making the following diagram commute:

(L,S) Sym(L,S)

Symmetric Frith frames

A Frith frame (L,S) is symmetric if S is a Boolean algebra.

Symmetric Frith frames form a full reflective subcategory of Frith.

That is, for every Frith frame (L,S), there exists a symmetric Frith frame

Sym(L,S) = (C(L,S), 〈S〉BA),

called the symmetrization of (L,S), such that for every h : (L,S)→ (M,B) with

(M,B) symmetric there is a unique h making the following diagram commute:

(L,S) Sym(L,S)

Completion of Frith frames

I A symmetric Frith frame (L,B) is complete if every dense surjection1

(M,C ) (L,B) with (M,C ) symmetric is an isomorphism.

I A Frith frame (L,S) is complete provided Sym(L,S) is complete.

A Frith frame (L,S) is complete if and only if L = Idl(S).

Corollary

The categories of coherent frames and of complete Frith frames are

isomorphic.

1h : (M,C) (L,B) is a dense surjection if (h(a) = 0 =⇒ a = 0) and h[C ] = B.

Corollary

isomorphic.

Corollary

isomorphic.

Corollary

isomorphic.

Proof’s idea:

I A completion of (L,S) is a dense surjection (M,T ) (L,S) with

(M,T ) complete.

I We have a dense surjection c : (Idl(S), S) (L,S), J 7→∨

I To show that (Idl(S),S) is complete:(every dense surjection (M,B) Sym(Idl(S),S) is an isomorphism)

Sym(Idl(S),S) (M,B)

Sym(L,S)Sym(L Idl(S),S)

(Idl(S),S)

(L,S)(L Idl(S),S)

cc id cc id

Therefore, h is one-one, thus an isomorphism.

Proof’s idea:

(M,T ) complete.

I We have a dense surjection c : (Idl(S), S) (L, S), J 7→∨J.

I To show that (Idl(S),S) is complete:(every dense surjection (M,B) Sym(Idl(S),S) is an isomorphism)

Sym(Idl(S),S) (M,B)

(Idl(S),S)

(L,S)(L Idl(S),S)

cc id cc id

Proof’s idea:

(M,T ) complete.

I To show that (Idl(S), S) is complete:(every dense surjection (M,B) Sym(Idl(S),S) is an isomorphism)

Sym(Idl(S),S) (M,B)

(Idl(S),S)

(L,S)(L Idl(S), S)

cc id cc id

Proof’s idea:

(M,T ) complete.

Sym(Idl(S),S) (M,B)

(Idl(S),S)

(L Idl(S), S)

c id cc id

Proof’s idea:

(M,T ) complete.

Sym(Idl(S), S)

Sym(L,S)

Sym(L Idl(S),S)

(Idl(S),S)

(L Idl(S), S)

Proof’s idea:

(M,T ) complete.

Sym(Idl(S), S) (M,B)

Sym(L,S)

Sym(L Idl(S),S)

(Idl(S),S)

(L Idl(S), S)

Proof’s idea:

(M,T ) complete.

Sym(L,S)

Sym(L Idl(S),S)

(Idl(S),S)

(L Idl(S), S)

Proof’s idea:

(M,T ) complete.

Sym(L,S)

Sym(L Idl(S),S)

(Idl(S),S)

(L Idl(S), S)

Completion of Pervin spaces

Theorem (Gehrke, Grigorieff, Pin, 2010; Pin, 2017)

The categories of spectral spaces and of complete T0 Pervin spaces are

isomorphic.

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

CPervin CFrithop

≡ ≡

The bitopological point of view

Bitopological spaces and biframes

A bitopological space is a triple (X , τ1, τ2), where τi is a topology on X .

A biframe is a triple (L, L1, L2) of frames st Li ≤ L and L = 〈L1 ∪L2〉Frm.

I Ω : biTop→ biFrmop

Ω(X , τ1, τ2) := (τ1 ∨ τ2, τ1, τ2)

I Σ : biFrmop → biTop

Σ(L, L1, L2) := (Σ(L), a | a ∈ L1, a | a ∈ L2)

biTop biFrmop

Top Frmop

Bitopological spaces and biframes

A bitopological space is a triple (X , τ1, τ2), where τi is a topology on X .

A biframe is a triple (L, L1, L2) of frames st Li ≤ L and L = 〈L1 ∪L2〉Frm.

I Ω : biTop→ biFrmop

Ω(X , τ1, τ2) := (τ1 ∨ τ2, τ1, τ2)

I Σ : biFrmop → biTop

Σ(L, L1, L2) := (Σ(L), a | a ∈ L1, a | a ∈ L2)

biTop biFrmop

Top Frmop

Pairwise Stone spaces are dual to bounded distributive lattices

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

CPervin CFrithop

≡ ≡

biTopbiTop biFrmop

PStone

Bezhanishvili et al.

Pairwise Stone spaces are dual to bounded distributive lattices

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

CPervin CFrithop

≡ ≡

biTopbiTop biFrmop

PStone

The Skula functor

If (X ,S) is a Pervin space, (X , 〈S〉Top, 〈Uc | U ∈ S〉Top) is a bispace.

This defines a full embedding

SkPervin : Pervin → biTop

Proposition (Bezhanishvili et al., 2010)

A bitopological space is a pairwise Stone space (i.e., pairwise compact,

pairwise Hausdorff, and pairwise 0-dimensional) if and only if it is of the

form SkPervin(X ,S) for some complete T0 Pervin space (X ,S).

The bitopological space (X , τ1, τ2) is pairwise...

− compact if every cover of τ1 ∪ τ2 has a finite refinement;

− Hausdorff if for every x 6= y , ∃Ui ∈ τi disjoint : x ∈ Uk , y ∈ U`, k, ` = 1, 2;

− 0-dimensional if U ∈ τk | Uc ∈ τ` is a basis for τk , where k, ` = 1, 2.

The Skula functor

If (X ,S) is a Pervin space, (X , 〈S〉Top, 〈Uc | U ∈ S〉Top) is a bispace.

SkPervin : Pervin → biTop

Proposition (Bezhanishvili et al., 2010)

A bitopological space is a pairwise Stone space (i.e., pairwise compact,

pairwise Hausdorff, and pairwise 0-dimensional) if and only if it is of the

form SkPervin(X ,S) for some complete T0 Pervin space (X ,S).

The bitopological space (X , τ1, τ2) is pairwise...

− compact if every cover of τ1 ∪ τ2 has a finite refinement;

− Hausdorff if for every x 6= y , ∃Ui ∈ τi disjoint : x ∈ Uk , y ∈ U`, k, ` = 1, 2;

− 0-dimensional if U ∈ τk | Uc ∈ τ` is a basis for τk , where k, ` = 1, 2.C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 19

The Skula functor

If (L,S) is a Frith frame, (C(L, S), L, 〈sc | s ∈ S〉Frm) is a biframe.

SkFrith : Frith → biFrm

A biframe is compact and 0-dimensional if and only if it is of the form

SkFrith(L,S) for some complete Frith frame (L,S).

The biframe (L, L1, L2) is:

− compact if L is compact;

− 0-dimensional if Li = 〈a ∈ Li complemented | ¬a ∈ Lj〉Frm with i , j = 1, 2.

The Skula functor

If (L,S) is a Frith frame, (C(L, S), L, 〈sc | s ∈ S〉Frm) is a biframe.

SkFrith : Frith → biFrm

A biframe is compact and 0-dimensional if and only if it is of the form

SkFrith(L,S) for some complete Frith frame (L,S).

The biframe (L, L1, L2) is:

− compact if L is compact;

− 0-dimensional if Li = 〈a ∈ Li complemented | ¬a ∈ Lj〉Frm with i , j = 1, 2.

The Skula functor

The following square commutes up to natural isomorphism.

Frithop biFrmop

Pervin biTop

SkFrith

SkPervin

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

CPervin CFrithop

≡ ≡

biTop biFrmop

SkPervin SkFr

PStone biFrmopKZ

SkPervin

SkFrith

Thank you for your attention!

Top Frmop

Spec CohFrmop

notfull

∼=Σ

DLatop

∼=Stone

Priestley

Pervin Frithop

CPervin CFrithop

≡ ≡

biTop biFrmop

SkPervin SkFr

PStone biFrmopKZ

SkPervin

SkFrith

Thank you for your attention!

extending stone-priestley duality along full embeddings

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