Natural Duality and Bitopology M. Andrew Moshier Chapman University - - PowerPoint PPT Presentation

Natural Duality and Bitopology

• M. Andrew Moshier

Chapman University

August 2018

Moshier (Chapman) BLAST 2018 1 / 73

An Alternative Title

Moshier (Chapman) BLAST 2018 2 / 73

An Alternative Title

Moshier (Chapman) BLAST 2018 2 / 73

Outline

1

Motivation for point-free bitopology Why think bitopologically? Why think point-free? The general idea

2

Some background

3

The natural adjunction of bispaces and d-frames

4

Alternative formulation: skew frames with auxiliary relation

5

Bitopological separation and compactness Regularity, complete regularity and normality Compactness

6

Examples and applications

Moshier (Chapman) BLAST 2018 3 / 73

Motivation for point-free bitopology

Outline

1

Motivation for point-free bitopology Why think bitopologically? Why think point-free? The general idea

2

Some background

3

The natural adjunction of bispaces and d-frames

4

Alternative formulation: skew frames with auxiliary relation

5

Bitopological separation and compactness Regularity, complete regularity and normality Compactness

6

Examples and applications

Moshier (Chapman) BLAST 2018 4 / 73

Motivation for point-free bitopology Why think bitopologically?

The reals numbers Upper and lower reals The Euclidean topology on R is the join of two natural order-theoretic topologies. τl = lower opens τu = upper opens

Moshier (Chapman) BLAST 2018 5 / 73

Motivation for point-free bitopology Why think bitopologically?

The reals numbers Upper and lower reals The Euclidean topology on R is the join of two natural order-theoretic topologies. τl = lower opens τu = upper opens Euclidean and compact support reals The Euclidean topology has its de Groot dual: τE = standard Euclidean opens τ ∗

E = ∅ and all complements of compact sets

The latter plays a role in analysis when functions of compact support are considered.

Moshier (Chapman) BLAST 2018 5 / 73

Motivation for point-free bitopology Why think bitopologically?

General topologies Upper and lower Vietoris For a family F of subsets of a space X, define ✷U = {F ∈ F | F ⊆ U} ✸U = {F ∈ F | U ∩ F = ∅} These generate two closely related topologies on the “hyperspace” F.

Moshier (Chapman) BLAST 2018 6 / 73

Motivation for point-free bitopology Why think bitopologically?

General topologies Upper and lower Vietoris For a family F of subsets of a space X, define ✷U = {F ∈ F | F ⊆ U} ✸U = {F ∈ F | U ∩ F = ∅} These generate two closely related topologies on the “hyperspace” F. Domains Everyone is familiar with the Scott topology on a dcpo. The Lawson dual is analogous to the De Groot dual: λ = topology generated by complements of compact upper sets Under nice order-theoretic conditions, the Lawson dual determines the Scott topology.

Moshier (Chapman) BLAST 2018 6 / 73

Motivation for point-free bitopology Why think bitopologically?

Natural duality for ordered algebras When an algebra is equipped with order so that operations are monotonic, there are two candidates for duals: Identify well-behaved lower sets (e.g, prime ideals in the case of distributive lattices). Identify well-behaved upper sets (e.g, all filters in the case of meet semilattices). The choice is in a certain sense conventional. Keeping both candidates in mind is often helpful. The usual way of constructing topological duals takes the join of these two (so the order is extra data).

Moshier (Chapman) BLAST 2018 7 / 73

Motivation for point-free bitopology Why think bitopologically?

Bispaces Definition On the principle that two can live as cheaply as one: A bispace is a set with two topologies living on it: (X, τ−, τ+) We call τ− the negative and τ+ the positive topology. A bicontinuous function between bispaces is simply a function that is continuous with respect to both the positive and negative topologies.

Moshier (Chapman) BLAST 2018 8 / 73

Motivation for point-free bitopology Why think bitopologically?

Bispaces Definition On the principle that two can live as cheaply as one: A bispace is a set with two topologies living on it: (X, τ−, τ+) We call τ− the negative and τ+ the positive topology. A bicontinuous function between bispaces is simply a function that is continuous with respect to both the positive and negative topologies. Why this is a not an interesting definition Having two unrelated topologies on a set is not very useful. It becomes useful when τ− and τ+ are related (as in all our earlier examples).

Moshier (Chapman) BLAST 2018 8 / 73

Motivation for point-free bitopology Why think bitopologically?

A quick review of Priestley duality Definition (Spectrum of a distributive lattice) For bounded distributive lattice L,

spec0(L) = all homomorphisms into 2

has two natural topologies τ− and τ+, generated by sets ϕ−

a = {p: L → 2 | p(a) = 0}

ϕ+

a = {p: L → 2 | p(a) = 1}

Moshier (Chapman) BLAST 2018 9 / 73

Motivation for point-free bitopology Why think bitopologically?

A quick review of Priestley duality Definition (Spectrum of a distributive lattice) For bounded distributive lattice L,

spec0(L) = all homomorphisms into 2

has two natural topologies τ− and τ+, generated by sets ϕ−

a = {p: L → 2 | p(a) = 0}

ϕ+

a = {p: L → 2 | p(a) = 1}

Notes: each is spectral (sober and generated by compact opens, which are closed under finite intersection). Closed sets of τ± are precisely the compact saturated subsets of τ∓. Joining the topologies yields a Stone space.

Moshier (Chapman) BLAST 2018 9 / 73

Motivation for point-free bitopology Why think bitopologically?

A quick review of Priestley duality The Priestley space

specP(L) = (spec0(L), τ− ⊔ τ+, ⊑−)

satisfies (spec0(L), τ− ⊔ τ+) is a Stone space (compact, 0-d); ⊑− is a partial order; If for every upper clopen u, x ∈ u implies y ∈ u, then x ⊑− y (this is Priestley separation). Ordered spaces that arise this way are characterized (up to homeomorphism) by these conditions.

Moshier (Chapman) BLAST 2018 10 / 73

Motivation for point-free bitopology Why think bitopologically?

A quick review of Priestley duality For h: L → M, the function specP(h): spec0(M) → spec0(L) is defined by

specP(h)(p) = p ◦ h.

Then specP(h) is continuous and order preserving from specP(M) to

specP(L).

But Upper clopen means compact open in τ−. Continuous and order preserving means continuous with respect τ− and τ+, separately So Priestley duality is naturally reformulated in terms of bitopological spaces.

Moshier (Chapman) BLAST 2018 11 / 73

Motivation for point-free bitopology Why think bitopologically?

Priestley Duality as Bitopology Definition (Priestley bispace) A Priestley bispace is a bispace (X, τ−, τ+) for which τ± is spectral; Compact saturated in τ± = closed in τ∓;

Moshier (Chapman) BLAST 2018 12 / 73

Motivation for point-free bitopology Why think bitopologically?

Priestley Duality as Bitopology Definition (Priestley bispace) A Priestley bispace is a bispace (X, τ−, τ+) for which τ± is spectral; Compact saturated in τ± = closed in τ∓; Priestley duality DLat PrBiSpop

specP KOP

≡

Moshier (Chapman) BLAST 2018 12 / 73

Motivation for point-free bitopology Why think point-free?

Frames are generalized spaces Three standard “reasons” for point-free topology. Many arguments in spaces are simplified because they do not really mention points anyway. Some even avoid AC.

Moshier (Chapman) BLAST 2018 13 / 73

Motivation for point-free bitopology Why think point-free?

Frames are generalized spaces Three standard “reasons” for point-free topology. Many arguments in spaces are simplified because they do not really mention points anyway. Some even avoid AC. Frames are algebraic objects (with an infinitary operation). Free

• bjects, objects specified by generators and relations exist. The

frame of the reals is a good example.

Moshier (Chapman) BLAST 2018 13 / 73

Motivation for point-free bitopology Why think point-free?

Frames are generalized spaces Three standard “reasons” for point-free topology. Many arguments in spaces are simplified because they do not really mention points anyway. Some even avoid AC. Frames are algebraic objects (with an infinitary operation). Free

• bjects, objects specified by generators and relations exist. The

frame of the reals is a good example. Frames genuinely extend the theory of topological spaces.

(Co)products are not the same; Subspaces are more numerous, even in a spatial frame.

The most striking example is Isbell’s Density Theorem: every point-free space as a smallest dense subspace.

Moshier (Chapman) BLAST 2018 13 / 73

Motivation for point-free bitopology Why think point-free?

Pointed spaces versus point-free spaces The functors pt and Ω The dual adjunction between frames and spaces is mediated by the two element frame 2 = •

• and Sierpi´

nski space space S = •

• :

Moshier (Chapman) BLAST 2018 14 / 73

Motivation for point-free bitopology Why think point-free?

Pointed spaces versus point-free spaces The functors pt and Ω The dual adjunction between frames and spaces is mediated by the two element frame 2 = •

• and Sierpi´

nski space space S = •

• :

pt(L) = [L → •

• ]

with topology induced from ( •

• )|L|

Ω(X) = [X → •

• ]

with frame order induced from ( •

• )|X|

Moshier (Chapman) BLAST 2018 14 / 73

Motivation for point-free bitopology Why think point-free?

Pointed spaces versus point-free spaces The functors pt and Ω The dual adjunction between frames and spaces is mediated by the two element frame 2 = •

• and Sierpi´

nski space space S = •

• :

pt(L) = [L → •

• ]

with topology induced from ( •

• )|L|

Ω(X) = [X → •

• ]

with frame order induced from ( •

• )|X|

Notes: Ω(X) is essentially sending the space (X, τ) to τ — so it “forgets” the points, but retains the topology.

pt(L) is essentially sending L to its principle prime ideals.

Moshier (Chapman) BLAST 2018 14 / 73

Motivation for point-free bitopology Why think point-free?

Pointed spaces and point-free spaces The pointed space/point-free space adjunction Frm Spop

pt

Ω ⊤

Moshier (Chapman) BLAST 2018 15 / 73

Motivation for point-free bitopology Why think point-free?

Pointed spaces and point-free spaces The pointed space/point-free space adjunction Frm Spop

pt

Ω ⊤ Remarks: When regarding frames as point-free spaces, it is reasonable to consider the category of locales: Loc = Frmop. In particular, “subframe” and “sublocale” are very different. Sublocale is the right notion of point-free subspace. Convention: We write ⊑, ⊓, , ⊥, ⊤ for frames.

Moshier (Chapman) BLAST 2018 15 / 73

Motivation for point-free bitopology The general idea

Toward point-free bispaces What we want A point-free bispace ought to look like this: (L−, L+, something) where L− and L+ are frames (point-free topologies); “something” is data describing how L− and L+ are actually talking about the same underlying point-free space. Continuity Evidently, the morphisms will be pairs of frame homomorphisms (f−, f+) that preserve “something”.

Moshier (Chapman) BLAST 2018 16 / 73

Motivation for point-free bitopology The general idea

Biframes: Maximal something Definition (Biframes) A biframe is a triple (L−, L+, L) so that L is also a frame — intended to encode the joint topology of L− and L+; hence L−, L+ are subframes of L; L is generated as a frame from L− and L+. Morphisms are frame homomorphims f : L → M that restrict to the positive and negative parts in the obvious way.

Moshier (Chapman) BLAST 2018 17 / 73

Motivation for point-free bitopology The general idea

Biframes: Maximal something Definition (Biframes) A biframe is a triple (L−, L+, L) so that L is also a frame — intended to encode the joint topology of L− and L+; hence L−, L+ are subframes of L; L is generated as a frame from L− and L+. Morphisms are frame homomorphims f : L → M that restrict to the positive and negative parts in the obvious way. The biframe adjunction BiFrm BiSpop

ptb

Ωb ⊤

Moshier (Chapman) BLAST 2018 17 / 73

Motivation for point-free bitopology The general idea

What’s wrong with biframes Pointed and point-free (uni)spaces Recall the adjunction between frames and spaces: Ω(X) = [X → •

• ]

pt(L) = [L → •

• ]

Moshier (Chapman) BLAST 2018 18 / 73

Motivation for point-free bitopology The general idea

What’s wrong with biframes Pointed and point-free (uni)spaces Recall the adjunction between frames and spaces: Ω(X) = [X → •

• ]

pt(L) = [L → •

• ]

Theorem No set D can be equipped with a biframe and bispace structure so that Ωb(X) = [X → DBiSp]

ptb(L) = [L → DBiFrm]

Moshier (Chapman) BLAST 2018 18 / 73

Motivation for point-free bitopology The general idea

What’s wrong with biframes Pointed and point-free (uni)spaces Recall the adjunction between frames and spaces: Ω(X) = [X → •

• ]

pt(L) = [L → •

• ]

Theorem No set D can be equipped with a biframe and bispace structure so that Ωb(X) = [X → DBiSp]

ptb(L) = [L → DBiFrm]

Moral of the story: Biframes are unnatural (technically speaking).

Moshier (Chapman) BLAST 2018 18 / 73

Some background

Outline

1

Motivation for point-free bitopology Why think bitopologically? Why think point-free? The general idea

2

Some background

3

The natural adjunction of bispaces and d-frames

4

Alternative formulation: skew frames with auxiliary relation

5

Bitopological separation and compactness Regularity, complete regularity and normality Compactness

6

Examples and applications

Moshier (Chapman) BLAST 2018 19 / 73

Some background

Double lattices and disjointed lattices Definition (The categories DLat2 and DLat2) The category DLat2 consists of double distributive lattices: pairs

• f distributive lattices and pairs of lattice homomorphisms with no

additional structure.

Moshier (Chapman) BLAST 2018 20 / 73

Some background

Double lattices and disjointed lattices Definition (The categories DLat2 and DLat2) The category DLat2 consists of double distributive lattices: pairs

• f distributive lattices and pairs of lattice homomorphisms with no

additional structure. The category DLat2 consists of disjointed distributive lattices: Objects (L, f, t) where L is a distributive lattice and f, t ∈ L are complements of each other. Morphisms Lattice homomorphisms that preserve f and t.

Moshier (Chapman) BLAST 2018 20 / 73

Some background

Double lattices and disjointed lattices Definition (The categories DLat2 and DLat2) The category DLat2 consists of double distributive lattices: pairs

• f distributive lattices and pairs of lattice homomorphisms with no

additional structure. The category DLat2 consists of disjointed distributive lattices: Objects (L, f, t) where L is a distributive lattice and f, t ∈ L are complements of each other. Morphisms Lattice homomorphisms that preserve f and t. Theorem The categories DLat2 and DLat2 are equivalent.

Moshier (Chapman) BLAST 2018 20 / 73

Some background

Double lattices and disjointed lattices Definition (The categories DLat2 and DLat2) The category DLat2 consists of double distributive lattices: pairs

• f distributive lattices and pairs of lattice homomorphisms with no

additional structure. The category DLat2 consists of disjointed distributive lattices: Objects (L, f, t) where L is a distributive lattice and f, t ∈ L are complements of each other. Morphisms Lattice homomorphisms that preserve f and t. Theorem The categories DLat2 and DLat2 are equivalent. Proof: (L−, L+) → (L− × L+, (1, 0), (0, 1)) (L, f, t) → (↑ t, ↑ f)

Moshier (Chapman) BLAST 2018 20 / 73

Some background

Some conventions Double lattice elements In a double lattice (L−, L+), we write a, b, . . . for elements of L+; and ϕ, ψ, . . . for elements of L−. Disjointed lattice elements In a disjointed lattice (L, f, t), we write α, β, γ ∈ L α− = α ⊔ t α+ = α ⊔ f α ∨ β = (t ⊓ α) ⊔ (α ⊓ β) ⊔ (β ⊓ t) α ∧ β = (f ⊓ α) ⊔ (α ⊓ β) ⊔ (β ⊓ f)

Moshier (Chapman) BLAST 2018 21 / 73

Some background

Some conventions Double lattice elements In a double lattice (L−, L+), we write a, b, . . . for elements of L+; and ϕ, ψ, . . . for elements of L−. Disjointed lattice elements In a disjointed lattice (L, f, t), we write α, β, γ ∈ L α− = α ⊔ t — projection onto L− α+ = α ⊔ f — projection onto L+ α ∨ β = (t ⊓ α) ⊔ (α ⊓ β) ⊔ (β ⊓ t) α ∧ β = (f ⊓ α) ⊔ (α ⊓ β) ⊔ (β ⊓ f)

Moshier (Chapman) BLAST 2018 21 / 73

Some background

Some conventions Double lattice elements In a double lattice (L−, L+), we write a, b, . . . for elements of L+; and ϕ, ψ, . . . for elements of L−. Disjointed lattice elements In a disjointed lattice (L, f, t), we write α, β, γ ∈ L α− = α ⊔ t — projection onto L− α+ = α ⊔ f — projection onto L+ α ∨ β = (t ⊓ α) ⊔ (α ⊓ β) ⊔ (β ⊓ t) α ∧ β = (f ⊓ α) ⊔ (α ⊓ β) ⊔ (β ⊓ f) — explained by the following

Moshier (Chapman) BLAST 2018 21 / 73

Some background

The 45◦ lemma (two orders for the price of one) Lemma Suppose (L, ⊔, ⊓, ⊥, ⊤; f, t) is a bounded distributive lattice with complementary elements. Define x ∧ y = (f ⊓ x) ⊔ (x ⊓ y) ⊔ (y ⊓ f) x ∨ y = (t ⊓ x) ⊔ (x ⊓ y) ⊔ (y ⊓ t) Then (L, ∨, ∧, f, t; ⊥, ⊤) is also a bounded distributive lattice with complementary elements. Moreover, this construction is involutive. Hence, objects of DLat2 have two orders: the information order — the given order; the logical order — defined by ∧ and/or ∨.

Moshier (Chapman) BLAST 2018 22 / 73

Some background

The 45◦ lemma f t ⊤ ⊥ f t ⊤ ⊥ Useful because we can speak about both orders. The information order: ⊑; The logical order: ≤.

Moshier (Chapman) BLAST 2018 23 / 73

Some background

Auxiliary relations An idea that has come up in several talks (e.g., George Metcalfe’s tutorials) . . . Definition A lattice relation is a relation R ⊆ L × M between lattices satisfying: Weakening x ≤L x′ R y′ ≤M y implies x R y; Logical R is a sub-lattice of L × M:

Moshier (Chapman) BLAST 2018 24 / 73

Some background

Auxiliary relations An idea that has come up in several talks (e.g., George Metcalfe’s tutorials) . . . Definition A lattice relation is a relation R ⊆ L × M between lattices satisfying: Weakening x ≤L x′ R y′ ≤M y implies x R y; Logical R is a sub-lattice of L × M:

0 R z; x R z and y R z implies x ∨ y R z; x R 1; x R y and x R z implies x R y ∧ z;

A relation ≺ is interpolative if x ≺ z implies x ≺ y ≺ z for some y.

Moshier (Chapman) BLAST 2018 24 / 73

Some background

Auxiliary relations An idea that has come up in several talks (e.g., George Metcalfe’s tutorials) . . . Definition A lattice relation is a relation R ⊆ L × M between lattices satisfying: Weakening x ≤L x′ R y′ ≤M y implies x R y; Logical R is a sub-lattice of L × M:

0 R z; x R z and y R z implies x ∨ y R z; x R 1; x R y and x R z implies x R y ∧ z;

An auxiliary relation on L is a lattice relation ≺⊆ L × L contained in the order relation. [It is thus transitive.] A relation ≺ is interpolative if x ≺ z implies x ≺ y ≺ z for some y.

Moshier (Chapman) BLAST 2018 24 / 73

Some background

Based distributive lattices Definition A based distributive lattice is a distributive lattice with designated element (L, ∧, ∨, 0, 1, ⊥). In a based distributive lattice, x ⊓ y = (⊥ ∧ x) ∨ (x ∧ y) ∨ (y ∧ ⊥) defines a semilattice operation that distributes over ∧ and ∨, and ⊥ is least element with respect to ⊑.

Moshier (Chapman) BLAST 2018 25 / 73

Some background

Based distributive lattices Definition A based distributive lattice is a distributive lattice with designated element (L, ∧, ∨, 0, 1, ⊥). In a based distributive lattice, x ⊓ y = (⊥ ∧ x) ∨ (x ∧ y) ∨ (y ∧ ⊥) defines a semilattice operation that distributes over ∧ and ∨, and ⊥ is least element with respect to ⊑. Example

• 1

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Some background

Based distributive lattices Definition A based distributive lattice is a distributive lattice with designated element (L, ∧, ∨, 0, 1, ⊥). In a based distributive lattice, x ⊓ y = (⊥ ∧ x) ∨ (x ∧ y) ∨ (y ∧ ⊥) defines a semilattice operation that distributes over ∧ and ∨, and ⊥ is least element with respect to ⊑. Example

• 1

⊥

Moshier (Chapman) BLAST 2018 25 / 73

Some background

Based distributive lattices Definition A based distributive lattice is a distributive lattice with designated element (L, ∧, ∨, 0, 1, ⊥). In a based distributive lattice, x ⊓ y = (⊥ ∧ x) ∨ (x ∧ y) ∨ (y ∧ ⊥) defines a semilattice operation that distributes over ∧ and ∨, and ⊥ is least element with respect to ⊑. Example

• 1

⊥ − →

• • •
• ⊥

1

Moshier (Chapman) BLAST 2018 25 / 73

Some background

Skew frames Definition A skew frame is a based distributive lattice (L, ∧, ∨, 0, 1, ⊥) for which the induced order ⊑ is a dcpo;

Moshier (Chapman) BLAST 2018 26 / 73

Some background

Skew frames Definition A skew frame is a based distributive lattice (L, ∧, ∨, 0, 1, ⊥) for which the induced order ⊑ is a dcpo; ∧ and ∨ preserve ↑.

Moshier (Chapman) BLAST 2018 26 / 73

Some background

Skew frames Definition A skew frame is a based distributive lattice (L, ∧, ∨, 0, 1, ⊥) for which the induced order ⊑ is a dcpo; ∧ and ∨ preserve ↑. Remarks: In any based distributive lattice, if x and y are bounded in ⊑, they have a least upper bound. Hence a skew frame is conditionally complete in ⊑. A frame is precisely a skew frame in which 0 = ⊥.

Moshier (Chapman) BLAST 2018 26 / 73

Some background

Double frames Definition The category Frm2 (double frames) is the subcategory of DLat2 consisting of frames and pairs of frame homomorphisms. The category Frm2 (disjointed frames) is the subcategory of DLat2 consisting of frames (L, f, t) with complements and frame homomorphisms that preserve f and t.

Moshier (Chapman) BLAST 2018 27 / 73

Some background

Double frames Definition The category Frm2 (double frames) is the subcategory of DLat2 consisting of frames and pairs of frame homomorphisms. The category Frm2 (disjointed frames) is the subcategory of DLat2 consisting of frames (L, f, t) with complements and frame homomorphisms that preserve f and t. Remarks: The 45◦ lemma still obtains, so that a disjointed frame has the given frame order (⊑) and its derived logical order (≤). A disjointed frame is a complete (distributive) lattice order in the logical order, not necessarily a frame or a coframe. Auxiliary relations must be specified as pertaining to ⊑ or ≤. Both kinds will appear later.

Moshier (Chapman) BLAST 2018 27 / 73

The natural adjunction of bispaces and d-frames

Outline

1

Motivation for point-free bitopology Why think bitopologically? Why think point-free? The general idea

2

Some background

3

The natural adjunction of bispaces and d-frames

4

Alternative formulation: skew frames with auxiliary relation

5

Bitopological separation and compactness Regularity, complete regularity and normality Compactness

6

Examples and applications

Moshier (Chapman) BLAST 2018 28 / 73

The natural adjunction of bispaces and d-frames

d-lattices Definition A d-lattice is a double distributive lattice (L−, L+, con, tot) equipped with relations

con ⊆ L+ × L− — intended to mean ϕ and a are disjoint tot ⊆ L− × L+ — intended to mean a and ϕ cover

satisfying

con-↓ con is a lower set con-logic con is a sublattice of L− × L∂

+

tot-↑ tot is an upper set tot-logic tot is a sublattice of L∂

+ × L−

con-tot

a con; tot b implies a ⊑+ b ψ tot; con ϕ implies ψ ⊒ ϕ

Moshier (Chapman) BLAST 2018 29 / 73

The natural adjunction of bispaces and d-frames

d-lattices Definition A d-lattice is a double distributive lattice (L−, L+, con, tot) equipped with relations

con ⊆ L+ × L− — intended to mean ϕ and a are disjoint tot ⊆ L− × L+ — intended to mean a and ϕ cover

satisfying

con-↓ con is a lower set con-logic con is a sublattice of L− × L∂

+

a lattice relation

tot-↑ tot is an upper set tot-logic tot is a sublattice of L∂

+ × L−

a lattice relation

con-tot

a con; tot b implies a ⊑+ b ψ tot; con ϕ implies ψ ⊒ ϕ

Moshier (Chapman) BLAST 2018 29 / 73

The natural adjunction of bispaces and d-frames

con-↓ and tot-↑

a ϕ

Moshier (Chapman) BLAST 2018 30 / 73

The natural adjunction of bispaces and d-frames

con-↓ and tot-↑

a ϕ b ψ

Moshier (Chapman) BLAST 2018 30 / 73

The natural adjunction of bispaces and d-frames

con-↓ and tot-↑

a ϕ b ψ ϕ a

Moshier (Chapman) BLAST 2018 30 / 73

The natural adjunction of bispaces and d-frames

con-↓ and tot-↑

a ϕ b ψ ϕ a ψ b

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The natural adjunction of bispaces and d-frames

”Logic” a ϕ

Moshier (Chapman) BLAST 2018 31 / 73

The natural adjunction of bispaces and d-frames

”Logic” b ψ

Moshier (Chapman) BLAST 2018 31 / 73

The natural adjunction of bispaces and d-frames

”Logic” a ∧ b ϕ ∨ ψ

Moshier (Chapman) BLAST 2018 31 / 73

The natural adjunction of bispaces and d-frames

”Logic” a ∧ b ϕ ∨ ψ Likewise for tot.

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The natural adjunction of bispaces and d-frames

con-tot

a ϕ a con ϕ

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The natural adjunction of bispaces and d-frames

con-tot

a ϕ b a con ϕ ϕ tot b

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The natural adjunction of bispaces and d-frames

con-tot

a ϕ b a con ϕ ϕ tot b So a ⊑ b

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The natural adjunction of bispaces and d-frames

The order dual of a d-lattice Definition For d-lattice L = (L−, L+, con, tot), define L∂ = (L∂

+, L∂ −, tot, con).

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The natural adjunction of bispaces and d-frames

The order dual of a d-lattice Definition For d-lattice L = (L−, L+, con, tot), define L∂ = (L∂

+, L∂ −, tot, con).

Because the axioms for con and tot are symmetric, this is clearly a d-lattice. (−)∂ extends to a functor by swapping component homomorphisms.

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The natural adjunction of bispaces and d-frames

The order dual of a d-lattice Definition For d-lattice L = (L−, L+, con, tot), define L∂ = (L∂

+, L∂ −, tot, con).

Because the axioms for con and tot are symmetric, this is clearly a d-lattice. (−)∂ extends to a functor by swapping component homomorphisms. Our next axiom defining d-frames breaks the symmetry, so L∂ of a d-frame is not a d-frame.

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The natural adjunction of bispaces and d-frames

The order dual of a d-lattice Definition For d-lattice L = (L−, L+, con, tot), define L∂ = (L∂

+, L∂ −, tot, con).

Because the axioms for con and tot are symmetric, this is clearly a d-lattice. (−)∂ extends to a functor by swapping component homomorphisms. Our next axiom defining d-frames breaks the symmetry, so L∂ of a d-frame is not a d-frame. We will use this when we consider ideal and filter completions.

Moshier (Chapman) BLAST 2018 33 / 73

The natural adjunction of bispaces and d-frames

d-frames Definition A d-frame is a d-lattice (L−, L+, con, tot) for which L− and L+ are frames, and

con-↓ con is a lower set con-logic con is a sublattice of L− × L∂

+;

tot-↑ tot is an upper set tot-logic tot is a sublattice of L+ × L∂

−

con-tot

a con; tot b implies a ⊑+ b ψ tot; con ϕ implies ψ ⊒− ϕ

con-↑ con is closed under directed joins

Moshier (Chapman) BLAST 2018 34 / 73

The natural adjunction of bispaces and d-frames

d-frames Definition A d-frame is a d-lattice (L−, L+, con, tot) for which L− and L+ are frames, and

con-↓ con is a lower set con-logic con is a sublattice of L− × L∂

+;

tot-↑ tot is an upper set tot-logic tot is a sublattice of L+ × L∂

−

con-tot

a con; tot b implies a ⊑+ b ψ tot; con ϕ implies ψ ⊒− ϕ

con-↑ con is closed under directed joins

We expected to formalize a point-free bispace as (L−, L+, something) with morphisms preserving “something.”

Moshier (Chapman) BLAST 2018 34 / 73

The natural adjunction of bispaces and d-frames

d-frames Definition A d-frame is a d-lattice (L−, L+, con, tot) for which L− and L+ are frames, and

con-↓ con is a lower set con-logic con is a sublattice of L− × L∂

+;

tot-↑ tot is an upper set tot-logic tot is a sublattice of L+ × L∂

−

con-tot

a con; tot b implies a ⊑+ b ψ tot; con ϕ implies ψ ⊒− ϕ

con-↑ con is closed under directed joins

We expected to formalize a point-free bispace as (L−, L+, something) with morphisms preserving “something.” So a d-frame homomorphism is two frame homomorphisms that, together, preserve con and tot.

Moshier (Chapman) BLAST 2018 34 / 73

The natural adjunction of bispaces and d-frames

con-↑

Moshier (Chapman) BLAST 2018 35 / 73

The natural adjunction of bispaces and d-frames

con-↑

Moshier (Chapman) BLAST 2018 35 / 73

The natural adjunction of bispaces and d-frames

con-↑

Moshier (Chapman) BLAST 2018 35 / 73

The natural adjunction of bispaces and d-frames

con-↑

Moshier (Chapman) BLAST 2018 35 / 73

The natural adjunction of bispaces and d-frames

con-↑

Moshier (Chapman) BLAST 2018 35 / 73

The natural adjunction of bispaces and d-frames

con-↑

Moshier (Chapman) BLAST 2018 35 / 73

The natural adjunction of bispaces and d-frames

con-↑

Moshier (Chapman) BLAST 2018 35 / 73

The natural adjunction of bispaces and d-frames

The dualizing object Definition Define the bispace S.S to be

• pen in τ−
• pen in τ+

Moshier (Chapman) BLAST 2018 36 / 73

The natural adjunction of bispaces and d-frames

The dualizing object Definition Define the bispace S.S to be

• pen in τ−
• pen in τ+

Definition Define the d-frame 2.2 to be

• tot

con

L− L+

Moshier (Chapman) BLAST 2018 36 / 73

The natural adjunction of bispaces and d-frames

The natural adjunction between pointed bispaces and pointfree bispaces Definition For bispace (X, τ−, τ+), Ωd(X) = all bicontinuous maps from X to S.S For d-frame (L−, L+, con, tot),

ptd(L) = all d-frame homomorphisms from L to 2.2

Moshier (Chapman) BLAST 2018 37 / 73

The natural adjunction of bispaces and d-frames

The natural adjunction between pointed bispaces and pointfree bispaces Definition For bispace (X, τ−, τ+), Ωd(X) = all bicontinuous maps from X to S.S For d-frame (L−, L+, con, tot),

ptd(L) = all d-frame homomorphisms from L to 2.2

Theorem d-Frmop BiSp

ptd

Ωd ⊤

Moshier (Chapman) BLAST 2018 37 / 73

The natural adjunction of bispaces and d-frames

So 2.2 (S.S) is the dualizer — what about truth values?

• Moshier (Chapman)

BLAST 2018 38 / 73

The natural adjunction of bispaces and d-frames

So 2.2 (S.S) is the dualizer — what about truth values?

• logical order ≤

Moshier (Chapman) BLAST 2018 38 / 73

The natural adjunction of bispaces and d-frames

So 2.2 (S.S) is the dualizer — what about truth values?

• logical order ≤

false true

Moshier (Chapman) BLAST 2018 38 / 73

The natural adjunction of bispaces and d-frames

So 2.2 (S.S) is the dualizer — what about truth values?

• logical order ≤

false true information order ⊑

Moshier (Chapman) BLAST 2018 38 / 73

The natural adjunction of bispaces and d-frames

So 2.2 (S.S) is the dualizer — what about truth values?

• logical order ≤

false true information order ⊑ No information

Moshier (Chapman) BLAST 2018 38 / 73

The natural adjunction of bispaces and d-frames

So 2.2 (S.S) is the dualizer — what about truth values?

• logical order ≤

false true information order ⊑ No information IDK

Moshier (Chapman) BLAST 2018 38 / 73

The natural adjunction of bispaces and d-frames

So 2.2 (S.S) is the dualizer — what about truth values?

• logical order ≤

false true information order ⊑ IDK What? True/False?

Moshier (Chapman) BLAST 2018 38 / 73

The natural adjunction of bispaces and d-frames

So 2.2 (S.S) is the dualizer — what about truth values?

• logical order ≤

false true information order ⊑ IDK What? True/False? WTF

Moshier (Chapman) BLAST 2018 38 / 73

The natural adjunction of bispaces and d-frames

Sub-d-frames Lemma For any d-frame L = (L−, L+, con, tot) and subframes, M− ⊆ L− and M+ ⊆ L+, (M−, M+, con ∩ (M+ × M−), tot ∩ (M− × M+)) is a d-frame, and the inclusion maps constitute a d-frame homomorphism.

Moshier (Chapman) BLAST 2018 39 / 73

The natural adjunction of bispaces and d-frames

Sub-d-frames Lemma For any d-frame L = (L−, L+, con, tot) and subframes, M− ⊆ L− and M+ ⊆ L+, (M−, M+, con ∩ (M+ × M−), tot ∩ (M− × M+)) is a d-frame, and the inclusion maps constitute a d-frame homomorphism. Proof. The result is clearly a sub-d-lattice. If {(ai, ϕi)}i ⊆ con ∩ (M+ × M−) is directed, then ↑aicon ↑ϕi. But these two joins are in M+ and M−, respectively.

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The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Let Idl(L) (Filt(L)) denote ideals (filters) of a distributive lattice — this is always a frame (the free frame over L, resp, L∂).

Moshier (Chapman) BLAST 2018 40 / 73

The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Let Idl(L) (Filt(L)) denote ideals (filters) of a distributive lattice — this is always a frame (the free frame over L, resp, L∂). Theorem Let L = (L−, L+, con, tot) be a d-lattice. For I ∈ Idl(L−), J ∈ Idl(L+), define I con∗ J ⇔ I × J ⊆ con J tot∗ I ⇔ (J × I) ∩ tot = ∅ Then L∗ = (Idl(L−), Idl(L+), con∗, tot∗) is the free d-frame over L.

Moshier (Chapman) BLAST 2018 40 / 73

The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Let Idl(L) (Filt(L)) denote ideals (filters) of a distributive lattice — this is always a frame (the free frame over L, resp, L∂). Theorem Let L = (L−, L+, con, tot) be a d-lattice. For I ∈ Idl(L−), J ∈ Idl(L+), define I con∗ J ⇔ I × J ⊆ con J tot∗ I ⇔ (J × I) ∩ tot = ∅ Then L∗ = (Idl(L−), Idl(L+), con∗, tot∗) is the free d-frame over L. Also, write Filt(L) for L∂

∗.

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The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Proof. Suppose M = (M−, M+, con′, tot′) is a d-frame and (h−, h+): L → M is a d-lattice homomorphism.

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The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Proof. Suppose M = (M−, M+, con′, tot′) is a d-frame and (h−, h+): L → M is a d-lattice homomorphism. h− and h+ are lattice homomorphisms.

Moshier (Chapman) BLAST 2018 41 / 73

The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Proof. Suppose M = (M−, M+, con′, tot′) is a d-frame and (h−, h+): L → M is a d-lattice homomorphism. h− and h+ are lattice homomorphisms. They lift uniquely to frame homomorphisms h†

± : Idl(L±) → M±.

Moshier (Chapman) BLAST 2018 41 / 73

The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Proof. Suppose M = (M−, M+, con′, tot′) is a d-frame and (h−, h+): L → M is a d-lattice homomorphism. h− and h+ are lattice homomorphisms. They lift uniquely to frame homomorphisms h†

± : Idl(L±) → M±.

Suppose I con∗ J. Then h−(a) con′ h+(ϕ) for each a ∈ I and ϕ ∈ J.

Moshier (Chapman) BLAST 2018 41 / 73

The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Proof. Suppose M = (M−, M+, con′, tot′) is a d-frame and (h−, h+): L → M is a d-lattice homomorphism. h− and h+ are lattice homomorphisms. They lift uniquely to frame homomorphisms h†

± : Idl(L±) → M±.

Suppose I con∗ J. Then h−(a) con′ h+(ϕ) for each a ∈ I and ϕ ∈ J. But I × J is directed and h†

+(I) = ↑ a∈I h+(a), and similarly for J.

Moshier (Chapman) BLAST 2018 41 / 73

The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Proof. Suppose M = (M−, M+, con′, tot′) is a d-frame and (h−, h+): L → M is a d-lattice homomorphism. h− and h+ are lattice homomorphisms. They lift uniquely to frame homomorphisms h†

± : Idl(L±) → M±.

Suppose I con∗ J. Then h−(a) con′ h+(ϕ) for each a ∈ I and ϕ ∈ J. But I × J is directed and h†

+(I) = ↑ a∈I h+(a), and similarly for J.

So h†

+(I) con′ h† −(J).

Moshier (Chapman) BLAST 2018 41 / 73

The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Proof. Suppose M = (M−, M+, con′, tot′) is a d-frame and (h−, h+): L → M is a d-lattice homomorphism. h− and h+ are lattice homomorphisms. They lift uniquely to frame homomorphisms h†

± : Idl(L±) → M±.

Suppose I con∗ J. Then h−(a) con′ h+(ϕ) for each a ∈ I and ϕ ∈ J. But I × J is directed and h†

+(I) = ↑ a∈I h+(a), and similarly for J.

So h†

+(I) con′ h† −(J).

Suppose J tot∗ I.

Moshier (Chapman) BLAST 2018 41 / 73

The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Proof. Suppose M = (M−, M+, con′, tot′) is a d-frame and (h−, h+): L → M is a d-lattice homomorphism. h− and h+ are lattice homomorphisms. They lift uniquely to frame homomorphisms h†

± : Idl(L±) → M±.

Suppose I con∗ J. Then h−(a) con′ h+(ϕ) for each a ∈ I and ϕ ∈ J. But I × J is directed and h†

+(I) = ↑ a∈I h+(a), and similarly for J.

So h†

+(I) con′ h† −(J).

Suppose J tot∗ I. Then ϕ tot a for some ϕ ∈ J and a ∈ I.

Moshier (Chapman) BLAST 2018 41 / 73

The natural adjunction of bispaces and d-frames

Ideal completion of a d-lattice Proof. Suppose M = (M−, M+, con′, tot′) is a d-frame and (h−, h+): L → M is a d-lattice homomorphism. h− and h+ are lattice homomorphisms. They lift uniquely to frame homomorphisms h†

± : Idl(L±) → M±.

Suppose I con∗ J. Then h−(a) con′ h+(ϕ) for each a ∈ I and ϕ ∈ J. But I × J is directed and h†

+(I) = ↑ a∈I h+(a), and similarly for J.

So h†

+(I) con′ h† −(J).

Suppose J tot∗ I. Then ϕ tot a for some ϕ ∈ J and a ∈ I. So h−(ϕ) tot′ h+(a)

Moshier (Chapman) BLAST 2018 41 / 73

Alternative formulation: skew frames with auxiliary relation

Outline

1

Motivation for point-free bitopology Why think bitopologically? Why think point-free? The general idea

2

Some background

3

The natural adjunction of bispaces and d-frames

4

Alternative formulation: skew frames with auxiliary relation

5

Bitopological separation and compactness Regularity, complete regularity and normality Compactness

6

Examples and applications

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Alternative formulation: skew frames with auxiliary relation

Trellises Recall skew frames are based distributive lattices in which the induced

• rder is directed complete.

Definition A trellis is a skew frame equipped with an auxiliary relation (L, ∧, ∨, t, f, ⊥, ≺) satisfying the additional property: emark: ⊥ ≺ ⊥ — by persistence, ⊥ ≺ ⊥ implies t ≺ f

Moshier (Chapman) BLAST 2018 43 / 73

Alternative formulation: skew frames with auxiliary relation

Trellises Recall skew frames are based distributive lattices in which the induced

• rder is directed complete.

Definition A trellis is a skew frame equipped with an auxiliary relation (L, ∧, ∨, t, f, ⊥, ≺) satisfying the additional property: Persistence α ⊒ α′ ≺ β′ ⊑ β implies α ≺ β emark: ⊥ ≺ ⊥ — by persistence, ⊥ ≺ ⊥ implies t ≺ f

Moshier (Chapman) BLAST 2018 43 / 73

Alternative formulation: skew frames with auxiliary relation

Trellises Recall skew frames are based distributive lattices in which the induced

• rder is directed complete.

Definition A trellis is a skew frame equipped with an auxiliary relation (L, ∧, ∨, t, f, ⊥, ≺) satisfying the additional property: Persistence α ⊒ α′ ≺ β′ ⊑ β implies α ≺ β A trellis homomorphism preserves all this structure (including directed joins with respect to ⊑) Remark: ⊥ ≺ ⊥ — by persistence, ⊥ ≺ ⊥ implies t ≺ f

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Alternative formulation: skew frames with auxiliary relation

d-frames and trellises From trellises to d-frames Given (L, ∧, ∨, t, f, ⊥, ≺), Let L− = [⊥, f] and L+ = [⊥, t] — both are frames. Define con ⊆ L+ × L− by a con ϕ if and only if {a, ϕ} is bounded with respect to ⊑. Define tot ⊆ L− × L+ by ϕ tot a if and only if ϕ ≺ a. The result is a d-frame.

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Alternative formulation: skew frames with auxiliary relation

d-frames and trellises From trellises to d-frames Given (L, ∧, ∨, t, f, ⊥, ≺), Let L− = [⊥, f] and L+ = [⊥, t] — both are frames. Define con ⊆ L+ × L− by a con ϕ if and only if {a, ϕ} is bounded with respect to ⊑. Define tot ⊆ L− × L+ by ϕ tot a if and only if ϕ ≺ a. The result is a d-frame. From d-frames to trellises Given a d-frame (L−, L+, con, tot), (con, ∧, ∨, (⊥+, ⊤−), (⊤+, ⊥−)) is a distributive lattice. (⊥+, ⊥−) as base determines a skew frame. Define (a, ϕ) ≺ (b, ψ) if and only if ϕ tot b. The result is a trellis.

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Alternative formulation: skew frames with auxiliary relation

Summarizing the story so far We have choices for the point-free counterparts of bitopological spaces.

Biframes take a maximalist approach, but do not have a natural dual adjunction. d-frames give us a natural dual adjunction by formalizing only point-free notions of “disjointness” and “covering”.

We have several equivalent ways to formulate d-frames.

(L−, L+, con, tot) — double frames with relations (L, f, t, con′, tot′) — disjointed frames with predicates (A, ∧, ∨, 1, 0, ⊥, ≺) — trellises (skew frames with persistent auxiliary relations)

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Bitopological separation and compactness

Outline

1

Motivation for point-free bitopology Why think bitopologically? Why think point-free? The general idea

2

Some background

3

The natural adjunction of bispaces and d-frames

4

Alternative formulation: skew frames with auxiliary relation

5

Bitopological separation and compactness Regularity, complete regularity and normality Compactness

6

Examples and applications

Moshier (Chapman) BLAST 2018 46 / 73

Bitopological separation and compactness

Sobriety and spatiality Won’t spend time on these, but briefly ... Definition (Sober bispaces) A bispace is sober if it is homeomorphic to some ptd(L) for a d-frame L. A d-frame is spatial if it is isomorphic to some Ωd(X) for a bispace X.

Moshier (Chapman) BLAST 2018 47 / 73

Bitopological separation and compactness

Sobriety and spatiality Won’t spend time on these, but briefly ... Definition (Sober bispaces) A bispace is sober if it is homeomorphic to some ptd(L) for a d-frame L. A d-frame is spatial if it is isomorphic to some Ωd(X) for a bispace X. Theorem A bispace X is sober if and only if it is homeomorphic to ptd(Ωd(X)) if and only if the unit of the adjunction η: X → ptd(Ωd(X)) is a bijection.

Moshier (Chapman) BLAST 2018 47 / 73

Bitopological separation and compactness

Sobriety and spatiality Won’t spend time on these, but briefly ... Definition (Sober bispaces) A bispace is sober if it is homeomorphic to some ptd(L) for a d-frame L. A d-frame is spatial if it is isomorphic to some Ωd(X) for a bispace X. Theorem A bispace X is sober if and only if it is homeomorphic to ptd(Ωd(X)) if and only if the unit of the adjunction η: X → ptd(Ωd(X)) is a bijection. Theorem A d-frame L is spatial if and only if L is isomorphic to Ωd(ptd(L)).

Moshier (Chapman) BLAST 2018 47 / 73

Bitopological separation and compactness

Points in a d-frame

Moshier (Chapman) BLAST 2018 48 / 73

Bitopological separation and compactness

Points in a d-frame A d-point

Moshier (Chapman) BLAST 2018 48 / 73

Bitopological separation and compactness

Points in a d-frame Not a d-point

Moshier (Chapman) BLAST 2018 48 / 73

Bitopological separation and compactness

Points in a d-frame Also not a d-point

Moshier (Chapman) BLAST 2018 48 / 73

Bitopological separation and compactness

The d-frame of the compact interval The d-frame L[0, 1]

• 1 0

1 ⊤− ⊤+

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Bitopological separation and compactness

The d-frame of the compact interval The d-frame L[0, 1]

• 1 0

1 ⊤− ⊤+ The d-point 3

4

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Bitopological separation and compactness Regularity, complete regularity and normality

Regularity There are weaker separation axioms, but regularity is a good place to start. Definition (Reminder) For opens u and v in a (uni)topological space X, u ⊳ v if and only if there is an open w so that u ∩ w = ∅ and w ∪ v = X. A space is regular if every open v is the union of opens u ⊳ v.

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Bitopological separation and compactness Regularity, complete regularity and normality

Regularity There are weaker separation axioms, but regularity is a good place to start. Definition (Reminder) For opens u and v in a (uni)topological space X, u ⊳ v if and only if there is an open w so that u ∩ w = ∅ and w ∪ v = X. A space is regular if every open v is the union of opens u ⊳ v. Definition (Regular d-frames) In a d-frame (L−, L+, con, tot) define ⊳− and ⊳+: ϕ ⊲− ψ if and only if ϕ tot a con ψ for some a ∈ L+. b ⊳+ a if and only if b con ϕ tot a for some ϕ ∈ L−. Then L is regular if every ϕ ∈ L− is the join of elements ψ ⊳− ϕ and likewise every a ∈ L+ is the join of elements b ⊳+ a.

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Bitopological separation and compactness Regularity, complete regularity and normality

Normality Definition (Reminder) A (uni)space is normal if for every two opens u and v such that u ∪ v = X, there are opens s and t so that u ∪ s = X, s ∩ t = ∅ and t ∪ v = X. This too translates to a simple property of d-frames.

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Bitopological separation and compactness Regularity, complete regularity and normality

Normality Definition (Reminder) A (uni)space is normal if for every two opens u and v such that u ∪ v = X, there are opens s and t so that u ∪ s = X, s ∩ t = ∅ and t ∪ v = X. This too translates to a simple property of d-frames. Definition (Normal d-frames) A d-frame (L−, L+, con, tot) is normal if and only if

tot = tot; con; tot

Moshier (Chapman) BLAST 2018 51 / 73

Bitopological separation and compactness Regularity, complete regularity and normality

Normality Definition (Reminder) A (uni)space is normal if for every two opens u and v such that u ∪ v = X, there are opens s and t so that u ∪ s = X, s ∩ t = ∅ and t ∪ v = X. This too translates to a simple property of d-frames. Definition (Normal d-frames) A d-frame (L−, L+, con, tot) is normal if and only if

tot = tot; con; tot

Note: Normality is a lattice property. It makes sense to speak about normal d-lattices.

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Bitopological separation and compactness Regularity, complete regularity and normality

Regularity and normality in trellises These definitions manifest in trellises as simple properties. Lemma A trellis corresponds to a regular d-frame if and only if for every α, ↑{β ⊓ γ | β ≺ α ≺ γ} Lemma A trellis corresponds to a normal d-frame if and only if ≺ is interpolative.

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Bitopological separation and compactness Compactness

Compactness Definition A d-frame (L−, L+, con, tot) is compact if and only if tot is open with respect to the Scott topology on L− × L+. Remarks: For any families A ⊆ L− and B ⊆ L+, if A tot B, then A0 tot B0 for some finite A0 ⊆ A and B0 ⊆ B. In bispaces, (X, τ−, τ+) is compact iff (X, τ− ∨ τ+) is compact in the usual topological sense (proof needs Alexander’s Subbase Lemma). L[0, 1] is compact; LR is not. A trellis is compact iff ≺ is logically compact: A ≺ B implies A0 ≺ B0 for finite A0 ⊆ A and B0 ⊆ B.

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Bitopological separation and compactness Compactness

Toward ˇ Cech-Stone compactification Recall that in topological spaces, the ˇ Cech-Stone compactification

• f a space (when it has a compactification) is its compact regular

reflection. So dually, in d-frames we are justified in looking for the compact regular co-reflection. In cases analogous to the spatial setting, we get a complete classification of all compactifications. In those cases, a largest one exists and it is the co-reflection.

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Bitopological separation and compactness Compactness

Nice ideals and filters Definition Let L = (L−, L+, con, tot) be a d-lattice. For ϕ ∈ L− and a ⊆ L+, define t−(ϕ) = {a ∈ L+ | ϕ tot a} c−(a) = {ϕ ∈ L− | a con ϕ} t+(a) = {ϕ ∈ L− | ϕ tot a} c+(ϕ) = {a ∈ L+ | a con ϕ} Lemma The maps t± are lattice homomorphisms L± → Filt(L∓). The maps c± are lattice homomorphisms L∓ → Idl(L±). Because of universal properties c± and t± extend to Filt(L∓) and

Idl(L±).

We will write ⇓± for c†

± ◦ t±.

Moshier (Chapman) BLAST 2018 55 / 73

Bitopological separation and compactness Compactness

Nice ideals and filters Lemma For d-lattice L, The frame endomorphisms c†

± ◦ t† ± are deflationary.

If L is normal, they are idempotent. Proof. ϕ ∈ c†

−(t† −(I)) if and only if ϕ ⊳− ψ for some ψ ∈ I.

ϕ ⊳− ψ implies ϕ ⊑− ψ by con-tot. So c†

−(t† −(I)) ⊆ I.

If L is normal, then (⊳±)2 = ⊳±. Consequently, normality implies the image of c†

± ◦ t† +± is a retract

• f Idl(L±).

This determines a sub-d-frame of L∗: L⋄ = (c†

−(t−(L−)), c† +(t+(L+)), con⋄, tot⋄).

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Bitopological separation and compactness Compactness

Compact regular co-reflection Theorem The assignment L → L⋄ is a functor from d-lattices to compact d-frames. And

1

if L is normal, L⋄ is regular

2

if L is a d-frame, then the assignments I → ↑I and J → ↑J constitute a d-frame homomorphism ǫL : L⋄ → L, natural in L.

3

If L is a regular d-frame, then the component maps of ǫL are surjections.

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Bitopological separation and compactness Compactness

Compact regular co-reflection Theorem The assignment L → L⋄ is a functor from d-lattices to compact d-frames. And

1

if L is normal, L⋄ is regular

2

if L is a d-frame, then the assignments I → ↑I and J → ↑J constitute a d-frame homomorphism ǫL : L⋄ → L, natural in L.

3

If L is a regular d-frame, then the component maps of ǫL are surjections. So if L is all three, then L⋄ is a compact regular d-frame and ǫL : L⋄ → L is a natural component-wise surjection.

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Bitopological separation and compactness Compactness

Proof: L⋄ is compact because J tot⋄ I simply means (J × I) ∩ tot = ∅ and directed joins of ideals are unions. Functoriality is straight-forward. I ∈ Idl⊳−(L−) is the directed union of ideals ⇓−ϕ for ϕ ∈ I. If L is normal, then ϕ ∈ I implies ⇓−ϕ ⊳− I. If L is a d-frame, the requested joins exist. The axiom con-↑ ensures that, together, they preserve con. Any maps that send ideals to upper bounds will preserve tot. By regularity, each ϕ ∈ L− is join of ⇓−ϕ.

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Bitopological separation and compactness Compactness

Compact regular co-reflection Lemma If L is a compact regular d-frame, then it is normal and ǫL is an isomorphism. Proof. The proof that L is normal mimics the familiar proof in spaces. So L⋄ is compact regular. The component maps of ǫL are surjections. By regularity, ϕ → ⇓−ϕ and a → ⇓+a are the inverses (as frame maps). ⇓± together preserve con because they ⇓−ϕ ⊆↓ ϕ and likewise for a ∈ L+. If a tot ϕ by regularity and compactness, there are elements b ⊳+ a and ψ ⊳− ϕ for which ψ tot b.

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Bitopological separation and compactness Compactness

ˇ Cech-Stone compactification We don’t yet have an honest ˇ Cech-Stone result because we have not settled on a precise notion of compactification. Definition A compactification of a d-frame L is a d-frame homomorphism f : L → M where M is compact regular, the components f− and f+ are surjective, and f is dense:

f+(a) con f−(ϕ) implies a con ϕ.

An easy consequence of the con-tot axioms is that f dense implies that its components are separately dense in the usual sense. Also easy: for regular, normal d-frame L, the natural morphism ǫL is dense.

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Bitopological separation and compactness Compactness

Complete regularity Definition In a d-frame: A scale from ϕ to ψ in L− is a map s from Q ∩ [0, 1] to L− for which s0 = ϕ, s1 = ψ and p < q implies sp ⊳− sq A scale from a to b in L+ is defined similarly Say ϕ is completely inside ψ (a is completely inside b) if there is a scale from ϕ to psi (a to b); Write ϕ ≪− ψ and a ≪+ b. The d-frame is completely regular if every ϕ and every a is the join of elements completely inside. There are other characterizations of complete regularity, but this is the easiest to use here.

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Bitopological separation and compactness Compactness

Some facts about complete regularity Lemma Any regular, normal d-frame is completely regular. Proof. By normality, ⊳± are interpolative, so with countable dependent choice, ⊳± and ≪± agree. By regularity, complete regularity now follows. Lemma If M is completely regular, and both components of f : M → L are surjective, then L is completely regular. Proof. For ϕ ∈ L−, ϕ = f−(ϕ′) for some ϕ′. But ϕ′ is the join of elements ψ ≪− ϕ′. And d-frame morphisms preserve ≪±.

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Bitopological separation and compactness Compactness

Normal co-reflection Before getting to the characterization of compactifications, the relations ≪± give us a little suprise. Theorem The category of normal d-frames is co-reflective in d-Frm. Proof. For (L−, L+, con, tot), define tot′ by ϕ tot′ a if and only there exist ψ and a so that b ≪+ a, ψ tot b and ψ ≪− ϕ. Since ≪± are included in the orders, the identity frame homomorphisms constitute a d-frame morphism from (L−, L+, con, tot′) to (L−, L+, con, tot). Clearly tot′ = tot′; con; tot′. So the result is normal.

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Bitopological separation and compactness Compactness

ˇ Cech-Stone compactification Theorem The category of compact regular d-frames is coreflective in the category of completely regular d-frames. Proof. For completely regular d-frame, its normal coreflection is regular (actually, still completely regular). So the compactification of the normalization of L is the desired d-frame.

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Bitopological separation and compactness Compactness

Some comments The normal coreflection step can be replaced by smaller normalizations using proximity relations. This leads to a range of compactifications — ˇ Cech-Stone being the largest. In (uni)spaces, the smallest compactification is X∗ — the one-point compactification. In completely regular d-frames, a smallest compactification always

• exists. What is it?

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Examples and applications

Outline

1

Motivation for point-free bitopology Why think bitopologically? Why think point-free? The general idea

2

Some background

3

The natural adjunction of bispaces and d-frames

4

Alternative formulation: skew frames with auxiliary relation

5

Bitopological separation and compactness Regularity, complete regularity and normality Compactness

6

Examples and applications

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Examples and applications

Compactifying the ordered reals Let LR be the d-frame of the reals. L− and L+ are the upper open and lower open topologies. (−∞, x) con (y, ∞) iff x ≤ y (y, ∞) tot (−∞, x) iff y < x This is already normal and regular. The d-frame LR ∅ ∅ R R

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Examples and applications

Compactifying the ordered reals Let LR be the d-frame of the reals. L− and L+ are the upper open and lower open topologies. (−∞, x) con (y, ∞) iff x ≤ y (y, ∞) tot (−∞, x) iff y < x This is already normal and regular. The d-frame LR ∅ ∅ R R The d-point 0 (−∞, 0) (0, ∞)

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Examples and applications

The largest compactification of LR The d-frame βLR {∅} {∅} {∅} ∪ {(∞, x) | x ∈ R} ∅ ∪ {(x, ∞) | x ∈ R}

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Examples and applications

The largest compactification of LR The d-frame βLR {∅} {∅} {∅} ∪ {(∞, x) | x ∈ R} ∅ ∪ {(x, ∞) | x ∈ R} {∅} ∪ {(x, ∞) | 0 < x}

• Moshier (Chapman)

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Examples and applications

The largest compactification of LR The d-frame βLR

• {∅} ∪ {(−∞, x) | x ∈ R} ∪ R

{∅} ∪ {(x, ∞) | x ∈ R} ∪ {∅} {∅} {∅} ∪ {(∞, x) | x ∈ R} ∅ ∪ {(x, ∞) | x ∈ R} {∅} ∪ {(x, ∞) | 0 < x}

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Examples and applications

Assembly of a frame Definition For a frame L, the assembly is N(L), the system of all quotients of L

• rdered by ⊇.

Theorem For any frame L, N(L) is a frame The map cL : L → N(L) defined by a → θ[0,a] is a frame homomorphism Each θ[0,a] is complemented in N(L) cL is universal with this property:

If f : L → M is a frame homomorphism so that f(a) is complemented, then there is a unique f † : N(L) → M so that f = f † ◦ cL.

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Examples and applications

An adjunction between Frm and d-Frm Definition For a frame L, define the symmetric d-frame S(L) = (L, L, conL, totL) x conL y iff x ∧ y = ⊥, and y totL x iff y ∨ x = ⊤. Definition For d-frame (L−, L+, con, tot), define the patch frame P(L−, L+, con, tot) = free frame generated by L− ⊎ L+ subject to relations already in L− and L+, a con ϕ implies a ∧ ϕ = ⊥, and ϕ tot a implies ϕ ∨ a = ⊤.

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Examples and applications

An adjunction between Frm and d-Frm Lemma P is left adjoint to S. Proof. This follows easily from the free construction of P. h: (L−, L+, con, tot) → S(M) determines two frame homomorphisms

h− : L− → M h+ : L+ → M

so that a con ϕ implies h−(a) ∧ h+(ϕ) = ⊥ and similarly for tot. Freeness of P(L−, L+, con, tot) lifts (h−, h+) to the required unique frame homomorphism h† : P(L−, L+, con, tot) → M.

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Examples and applications

Klinke’s d-frame Definition For a frame L, define K(L) = (Filt(L), L, con, tot) where a con F iff a ≤ b for all b ∈ F F tot a iff a ∈ F. Evidently, K is functorial because f : L → M lifts to a frame map

Filt(f): Filt(L) → Filt(M) so that (Filt(f), f) is d-frame morphism.

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Examples and applications

Klinke’s d-frame Definition For a frame L, define K(L) = (Filt(L), L, con, tot) where a con F iff a ≤ b for all b ∈ F F tot a iff a ∈ F. Evidently, K is functorial because f : L → M lifts to a frame map

Filt(f): Filt(L) → Filt(M) so that (Filt(f), f) is d-frame morphism.

Theorem For any frame L, the frame P(K(L)) satisfies the universal property of N(L).

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Examples and applications

Proof: The identity on L extends to a frame homomorphism c : L → P(K(L)) sending a to [a] — to get this smoothly, we really should have introduced some additional adjoint pairs of functors. In P(K(L)), [a] and [↑a] are complements because ↑a tot a and a con ↑a. Suppose f : L → M is a frame homomorphism so that each f(a) has a complement f(a)′ in M.

The function a → f(a)′ is a lattice homomorphism from L∂ to M. So this lifts to a frame homomorphism Φ: Filt(L) → M. Now check that (Φ, f) is a d-frame morphism from K(L) to S(M). Using the adjunction, we get the required frame homomorphism P(K(L)) → M.

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