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
Ω>Σ
Priestley DLatop
notfull
notfull
∼=
Pervin Frithop
Ω>Σ
Ω>Σ
CPervin CFrithop
full
full
∼= ∼=
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
Ω>Σ
Priestley DLatop
notfull
notfull
∼=
Pervin Frithop
Ω>Σ
Ω>Σ
CPervin CFrithop
full
full
∼= ∼=
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
Ω>Σ
Priestley DLatop
notfull
notfull
∼=
Pervin Frithop
Ω>Σ
Ω>Σ
CPervin CFrithop
full
full
∼= ∼=
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
Ω>Σ
Priestley DLatop
notfull
notfull
∼=
Pervin Frithop
Ω>Σ
Ω>Σ
CPervin CFrithop
full
full
∼= ∼=
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
Ω>Σ
Priestley DLatop
notfull
notfull
∼=
Pervin Frithop
Ω>Σ
Ω>Σ
CPervin CFrithop
full
full
∼= ∼=
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
The spatial-sober duality
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)
h−1
−−→ 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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 3
The spatial-sober duality
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)
h−1
−−→ 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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 3
The spatial-sober duality
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)
h−1
−−→ 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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 3
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
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4
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
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4
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
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4
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
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4
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
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4
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
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 5
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
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 5
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
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 5
Stone-Priestley duality for bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6
Stone-Priestley duality for bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6
Stone-Priestley duality for bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=
Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6
Stone-Priestley duality for bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6
Stone-Priestley duality for bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6
Stone-Priestley duality for bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6
Stone-Priestley duality for bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 7
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 8
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 8
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 8
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 9
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 9
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 9
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 )
h−1
−−→ Σ(L,S))
Pervin Frithop
Top Frmop
Ω>Σ
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 10
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 )
h−1
−−→ Σ(L,S))
Pervin Frithop
Top Frmop
Ω>Σ
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 10
Stone-Priestley duality for bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
Ω>Σ
CPervin CFrithop
full
full
≡ ≡
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 11
Stone-Priestley duality for bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
Ω>Σ
CPervin CFrithop
full
full
≡ ≡
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 11
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)
(M,B)
hh
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 12
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)
(M,B)
hh
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 12
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.
Theorem (B., Suarez)
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 13
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.
Theorem (B., Suarez)
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 13
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.
Theorem (B., Suarez)
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 13
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.
Theorem (B., Suarez)
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 13
Completion of Frith frames
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→∨
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)
Sym(L,S)Sym(L Idl(S),S)
h
h
(Idl(S),S)
(L,S)(L Idl(S),S)
cc id cc id
Therefore, h is one-one, thus an isomorphism.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14
Completion of Frith frames
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→∨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)
Sym(L,S)Sym(L Idl(S),S)
h
h
(Idl(S),S)
(L,S)(L Idl(S),S)
cc id cc id
Therefore, h is one-one, thus an isomorphism.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14
Completion of Frith frames
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→∨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)
Sym(L,S)Sym(L Idl(S),S)
h
h
(Idl(S),S)
(L,S)(L Idl(S), S)
cc id cc id
Therefore, h is one-one, thus an isomorphism.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14
Completion of Frith frames
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→∨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)
Sym(L,S)Sym(L Idl(S),S)
h
h
(Idl(S),S)
(L,S)
(L Idl(S), S)
c
c id cc id
Therefore, h is one-one, thus an isomorphism.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14
Completion of Frith frames
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→∨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)
Sym(L,S)
Sym(L Idl(S),S)
h
h
(Idl(S),S)
(L,S)
(L Idl(S), S)
c
c id
c
c id
Therefore, h is one-one, thus an isomorphism.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14
Completion of Frith frames
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→∨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)
Sym(L,S)
Sym(L Idl(S),S)
h
h
(Idl(S),S)
(L,S)
(L Idl(S), S)
c
c id
c
c id
Therefore, h is one-one, thus an isomorphism.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14
Completion of Frith frames
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→∨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)
Sym(L,S)
Sym(L Idl(S),S)
h
h
(Idl(S),S)
(L,S)
(L Idl(S), S)
c
c id
c
c id
Therefore, h is one-one, thus an isomorphism.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14
Completion of Frith frames
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→∨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)
Sym(L,S)
Sym(L Idl(S),S)
h
h
(Idl(S),S)
(L,S)
(L Idl(S), S)
c
c id
c
c id
Therefore, h is one-one, thus an isomorphism.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14
Completion of Pervin spaces
Theorem (Gehrke, Grigorieff, Pin, 2010; Pin, 2017)
The categories of spectral spaces and of complete T0 Pervin spaces are
isomorphic.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 15
Stone-Priestley duality for bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
Ω
Σ
CPervin CFrithop
full
full
≡ ≡
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 16
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
Ω>Σ
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 17
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
Ω>Σ
Ω>Σ
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 17
Pairwise Stone spaces are dual to bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
CPervin CFrithop
full
full
≡ ≡
biTopbiTop biFrmop
Ω>Σ
PStone
full
Bezhanishvili et al.
≡
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 18
Pairwise Stone spaces are dual to bounded distributive lattices
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
CPervin CFrithop
full
full
≡ ≡
biTopbiTop biFrmop
Ω>Σ
PStone
full
Bezhanishvili et al.
≡
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 18
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 19
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.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.
This defines a full embedding
SkFrith : Frith → biFrm
Proposition (B., Suarez)
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 20
The Skula functor
If (L,S) is a Frith frame, (C(L, S), L, 〈sc | s ∈ S〉Frm) is a biframe.
This defines a full embedding
SkFrith : Frith → biFrm
Proposition (B., Suarez)
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.
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 20
The Skula functor
Proposition (B., Suarez)
The following square commutes up to natural isomorphism.
Frithop biFrmop
Pervin biTop
SkFrith
ΣΣ
SkPervin
C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 21
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
CPervin CFrithop
full
full
≡ ≡
biTop biFrmop
Ω>Σ
SkPervin SkFr
ith
PStone biFrmopKZ
full
full
SkPervin
≡
Bezhanishvili et al.
≡
SkFrith
Thank you for your attention!
Top Frmop
Ω>Σ
Spec CohFrmop
notfull
notfull
Ω
∼=Σ
DLatop
∼=Stone
Priestley
∼=
Priestley
Pervin Frithop
Ω>Σ
CPervin CFrithop
full
full
≡ ≡
biTop biFrmop
Ω>Σ
SkPervin SkFr
ith
PStone biFrmopKZ
full
full
SkPervin
≡
Bezhanishvili et al.
≡
SkFrith
Thank you for your attention!