Extending Stone-Priestley Duality Along Full Embeddings

Extending Stone-Priestley duality along full embeddings

C´eliaBorlido based on joint work with A. L. Suarez

Centre for Mathematics, University of Coimbra

BLAST 2021 9 - 13 June Σ op Pervin > Frith Ω full full

CPervin CFrithop not full not full ∼ = op =∼ Priestley ∼= DLat

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

Σ op Top > Frm Ω Σ op Pervin > Frith Ω full full

CPervin CFrithop

=∼ =∼

Σ op Top > Frm Ω not full not full

op Priestley ∼= DLat Σ > Ω full full

CPervin CFrithop

=∼ =∼

Σ op Pervin > Frith Ω

Σ op Top > Frm Ω not full not full

op Priestley ∼= DLat Σ > Ω

Σ op Pervin > Frith Ω full full Σ op Top > Frm Ω CPervin CFrithop not full not full ∼ = op =∼ Priestley ∼= DLat Σ > Ω

Σ op Pervin > Frith Ω full full Σ op Top > Frm Ω CPervin CFrithop not full not full ∼ = op =∼ Priestley ∼= DLat The spatial-sober duality

Frames Topological spaces

A frame is a complete lattice L satisfying _ _ a ∧ b = (a ∧ b ). i i and continuous functions. i∈I

Frame homomorphisms preserve ﬁnite meets and arbitrary joins.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 2 op I Σ: Frm → Top

−1 Σ(L) := {c. p. ﬁlters of L} Σ(L −→h M) := (Σ(M) −−→h Σ(L))

ab := {F | a ∈ F }, a ∈ L

op We have an adjunction Ω : Top Frm : 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.

The spatial-sober duality

op I Ω: Top → Frm

−1 Ω(X ) := ({open subsets of X }, ⊆) Ω(X −→f Y ) := (Ω(Y ) −−→f Ω(X ))

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 3 op We have an adjunction Ω : Top Frm : 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.

The spatial-sober duality

op I Ω: Top → Frm

−1 Ω(X ) := ({open subsets of X }, ⊆) Ω(X −→f Y ) := (Ω(Y ) −−→f Ω(X ))

op I Σ: Frm → Top

−1 Σ(L) := {c. p. ﬁlters of L} Σ(L −→h M) := (Σ(M) −−→h Σ(L))

ab := {F | a ∈ F }, a ∈ L

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 3 The spatial-sober duality

op I Ω: Top → Frm

−1 Ω(X ) := ({open subsets of X }, ⊆) Ω(X −→f Y ) := (Ω(Y ) −−→f Ω(X ))

op I Σ: Frm → Top

−1 Σ(L) := {c. p. ﬁlters of L} Σ(L −→h M) := (Σ(M) −−→h Σ(L))

ab := {F | a ∈ F }, a ∈ L

op We have an adjunction Ω : Top Frm : 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 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→ hh[J]iIdl is 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

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.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4 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→ hh[J]iIdl is 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

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.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4 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

Bounded distributive lattices seen as coherent frames

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→ hh[J]iIdl is a coherent homomorphism.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4 These assignments deﬁne an equivalence of categories DLat =∼ CohFrm

Bounded distributive lattices seen as coherent frames

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→ hh[J]iIdl is 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.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 4 Bounded distributive lattices seen as coherent frames

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→ hh[J]iIdl is 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 maps are continuous functions such that the preimage of compact open subsets is again compact (and open).

op The adjunction Ω : Top Frm : Σ restricts and co-restricts to an equivalence

Spec =∼ CohFrmop

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.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 5 op The adjunction Ω : Top Frm : Σ restricts and co-restricts to an equivalence

Spec =∼ CohFrmop

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).

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 5 Spectral spaces

op The adjunction Ω : Top Frm : Σ restricts and co-restricts to an equivalence

Spec =∼ CohFrmop

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 5 Σ op Pervin > Frith Ω not full not full Σ op Spec ∼= CohFrm Ω =∼ Stone =∼ Priestley DLatop Priestley

Stone-Priestley duality for bounded distributive lattices

Σ op Top > Frm Ω

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6 Σ op Pervin > Frith Ω

=∼ Stone =∼ Priestley DLatop Priestley

Stone-Priestley duality for bounded distributive lattices

Σ op Top > Frm Ω not full not full Σ op Spec ∼= CohFrm Ω

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6 Σ op Pervin > Frith Ω

=∼ Stone Priestley Priestley

Stone-Priestley duality for bounded distributive lattices

Σ op Top > Frm Ω not full not full Σ op Spec ∼= CohFrm Ω =∼

DLatop

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6 Σ op Pervin > Frith Ω

=∼ Priestley Priestley

Stone-Priestley duality for bounded distributive lattices

Σ op Top > Frm Ω not full not full Σ op Spec ∼= CohFrm Ω Stone =∼ DLatop

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6 Σ op Pervin > Frith Ω

Priestley

Stone-Priestley duality for bounded distributive lattices

Σ op Top > Frm Ω not full not full Σ op Spec ∼= CohFrm Ω =∼ Stone =∼ Priestley DLatop

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6 Σ op Pervin > Frith Ω

Stone-Priestley duality for bounded distributive lattices

Σ op Top > Frm Ω not full not full Σ op Spec ∼= CohFrm Ω =∼ Stone =∼ Priestley DLatop Priestley

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 6 Stone-Priestley duality for bounded distributive lattices

Σ op Pervin > Frith Ω

Σ op Top > Frm Ω not full not full Σ op Spec ∼= CohFrm Ω =∼ Stone =∼ Priestley DLatop Priestley

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 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

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

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 8 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

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.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 8 Pervin spaces

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 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 of transitive and totally bounded quasi-uniform frames is equivalent to the category Frith of Frith frames.

Frith frames

A Frith frame is a pair (L, S), where L is a frame and S ⊆ L is a join-dense bounded sublattice.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 9 There is a full embedding Frm ,→ Frith, L 7→ (L, L).

Theorem (B., Suarez) The category of transitive and totally bounded quasi-uniform frames is equivalent to the category Frith of Frith frames.

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 .

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 9 Frith frames

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

Theorem (B., Suarez) The category of transitive and totally bounded quasi-uniform frames is equivalent to the category Frith of Frith frames.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 9 op I Σ: Frith → Pervin

Σ(L, S) := (Σ(L), {sb | s ∈ S})(sb := {F | s ∈ F }) −1 Σ((L, S) −→h (M, T )) := (Σ(M, T ) −−→h Σ(L, S))

Σ op Pervin > Frith Ω

Σ op Top > Frm Ω

The Frith-Pervin adjunction

op I Ω: Pervin → Frith

Ω(X , S) := (hSiFrm, S)

−1 Ω((X , S) −→f (Y , T )) := (Ω(Y , T ) −−→f Ω(X , S))

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 10 The Frith-Pervin adjunction

op I Ω: Pervin → Frith

Ω(X , S) := (hSiFrm, S)

−1 Ω((X , S) −→f (Y , T )) := (Ω(Y , T ) −−→f Ω(X , S))

op I Σ: Frith → Pervin

Σ(L, S) := (Σ(L), {sb | s ∈ S})(sb := {F | s ∈ F }) −1 Σ((L, S) −→h (M, T )) := (Σ(M, T ) −−→h Σ(L, S))

Σ op Pervin > Frith Ω

Σ op Top > Frm Ω C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 10 Σ > Ω full full

CPervin CFrithop

≡ ≡

Stone-Priestley duality for bounded distributive lattices

Σ op Pervin > Frith Ω

Σ op Top > Frm Ω not full not full Σ op Spec ∼= CohFrm Ω =∼ Stone =∼ Priestley DLatop Priestley

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 11 Σ > Ω

Stone-Priestley duality for bounded distributive lattices

Σ op Pervin > Frith Ω full full Σ op Top > Frm Ω CPervin CFrithop not full not full Σ ≡ op ≡ Spec ∼= CohFrm Ω =∼ Stone =∼ Priestley DLatop Priestley

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 11 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), hSiBA), 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)

h h

(M, B)

Symmetric Frith frames

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

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), hSiBA), 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)

h h

(M, B)

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 12 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.

Completion of Frith frames

1 I A symmetric Frith frame (L, B) is complete if every dense surjection (M, C) (L, B) with (M, C) symmetric is an isomorphism.

1 h :(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 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.

Completion of Frith frames

1 I A symmetric Frith frame (L, B) is complete if every dense surjection (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.

1 h :(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 Corollary The categories of coherent frames and of complete Frith frames are isomorphic.

Completion of Frith frames

1 I A symmetric Frith frame (L, B) is complete if every dense surjection (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).

1 h :(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

1 I A symmetric Frith frame (L, B) is complete if every dense surjection (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.

1 h :(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 h (Idl(S), S) Sym(Idl(S), S) b (M, B)

h ccid ccid (L Idl(L,(S), S) SymSym(L Idl(L,(S), S)

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

W 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)

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.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14 h (Idl(S), S) Sym(Idl(S), S) b (M, B)

h ccid ccid (L Idl(L,(S), S) SymSym(L Idl(L,(S), S)

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

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

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

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14 h (Idl(S), S) Sym(Idl(S), S) b (M, B)

h ccid ccid (L Idl(L,(S), S) SymSym(L Idl(L,(S), S)

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

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. W 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)

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14 h Sym(Idl(S), S) b (M, B)

h c id ccid (L Idl(S), S) SymSym(L Idl(L,(S), S)

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

Completion of Frith frames

(Idl(S), S)

(L, S)

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14 h b (M, B)

h c id c id (L Idl(S), S) Sym(L Idl(S), S)

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

Completion of Frith frames

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

c c (L, S) Sym(L, S)

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14 c id c id (L Idl(S), S) Sym(L Idl(S), S)

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

Completion of Frith frames

h c c (L, S) Sym(L, S)

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14 c c (L, S) Sym(L, S)

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

Completion of Frith frames

h c id c id (L Idl(S), S) Sym(L Idl(S), S)

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 14 c c (L, S) Sym(L, S)

Completion of Frith frames

h c id c id (L Idl(S), S) Sym(L Idl(S), S)

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

Σ op Pervin > Frith Ω full full Σ op Top > Frm Ω CPervin CFrithop not full not full Σ ≡ op ≡ Spec ∼= CohFrm Ω =∼ Stone =∼ Priestley DLatop Priestley

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 16 The bitopological point of view op I Ω: biTop → biFrm

Ω(X , τ1, τ2) := (τ1 ∨ τ2, τ1, τ2)

op I Σ: biFrm → biTop

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

Σ op biTop > biFrm Ω

Σ op Top > Frm Ω

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 = hL1 ∪ L2iFrm.

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 = hL1 ∪ L2iFrm.

op I Ω: biTop → biFrm

Ω(X , τ1, τ2) := (τ1 ∨ τ2, τ1, τ2)

op I Σ: biFrm → biTop

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

Σ op biTop > biFrm Ω

Σ op Top > Frm Ω

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 17 full

≡ et al. PStone Bezhanishvili

Pairwise Stone spaces are dual to bounded distributive lattices

Σ op biTop > biFrm Ω

Σ op Pervin > Frith Ω full full Σ op Top > Frm Ω CPervin CFrithop not full not full Σ ≡ op ≡ Spec ∼= CohFrm Ω =∼ Stone =∼ Priestley DLatop Priestley

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 18 Pairwise Stone spaces are dual to bounded distributive lattices

Σ op biTop > biFrm Ω

Σ op Pervin > Frith Ω full full full Σ op Top > Frm Ω CPervin CFrithop not full not full Σ ≡ op ≡ Spec ∼= CohFrm ≡ Ω et al. =∼ Stone =∼ PStone Bezhanishvili Priestley DLatop Priestley

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 18 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}; c − 0-dimensional if {U ∈ τk | U ∈ τ`} is a basis for τk , where {k, `} = {1, 2}.

The Skula functor

c If (X , S) is a Pervin space, (X , hSiTop, h{U | U ∈ S}iTop) is a bispace. This deﬁnes a full embedding

SkPervin : Pervin ,→ biTop

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 19 The Skula functor

c If (X , S) is a Pervin space, (X , hSiTop, h{U | U ∈ S}iTop) 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}; c − 0-dimensional if {U ∈ τk | U ∈ τ`} is a basis for τk , where {k, `} = {1, 2}.

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 19 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 = h{a ∈ Li complemented | ¬a ∈ Lj }iFrm with {i, j} = {1, 2}.

The Skula functor

c If (L, S) is a Frith frame, (C(L, S), L, h{s | s ∈ S}iFrm) is a biframe. This deﬁnes a full embedding

SkFrith : Frith ,→ biFrm

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 20 The Skula functor

c If (L, S) is a Frith frame, (C(L, S), L, h{s | s ∈ S}iFrm) 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 = h{a ∈ Li complemented | ¬a ∈ Lj }iFrm 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.

SkFrith Frithop biFrmop

Σ Σ

Pervin biTop SkPervin

C. Borlido (CMUC) Extending Stone-Priestley duality along full embeddings BLAST 2021 21 Thank you for your attention!

Σ op biTop > biFrm Sk Ω Pervin Frith Sk Σ op Pervin > Frith Ω full full full full Σ op Top > Frm Ω CPervin CFrithop

not full not full Sk Σ Frith Pervin ≡ op ≡ Sk Spec ∼= CohFrm ≡ Ω ≡ et al. ∼ ∼ = Stone = op PStone Bezhanishvili biFrmKZ Priestley DLatop Priestley Σ op biTop > biFrm Sk Ω Pervin Frith Sk Σ op Pervin > Frith Ω full full full full Σ op Top > Frm Ω CPervin CFrithop

not full not full Sk Σ Frith Pervin ≡ op ≡ Sk Spec ∼= CohFrm ≡ Ω ≡ et al. ∼ ∼ = Stone = op PStone Bezhanishvili biFrmKZ Priestley DLatop Priestley

Thank you for your attention!