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Applications of entailments: de Groot duality Tatsuji Kawai Japan - - PowerPoint PPT Presentation
Applications of entailments: de Groot duality Tatsuji Kawai Japan - - PowerPoint PPT Presentation
Applications of entailments: de Groot duality Tatsuji Kawai Japan Advanced Institute of Science and Technology Workshop DOMAINS 8 July 2018, Oxford 1 De Groot duality A topological space is stably compact if it is sober, locally compact, and
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Aim
We give a point-free (and constructive) account of de Groot duality. A locale X is stably compact if the frame Ω(X) is a continuous lattice and the set
։ x
def
=
- x′ ∈ X | x ≪ x′
is a filter for each x ∈ X. Stably compact locales are the Stone dual of stably compact spaces through the equivalence SoberSpa ∼
= SpatialLoc.
The de Groot dual X△ of a stably compact locale X is the frame of Scott open filters on Ω(X) (cf. Escardó 2000).
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I tried to reconstruct the dualities due to Goubault-Larrecq in the point-free setting, and got a couple of results, e.g. PU(X)d ∼
= PL(Xd).
But I got stuck . . .
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Plan
- 1. Strong proximity lattices
- 2. De Groot duality
- 3. Applications
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Stably compact locales
Proposition A locale is stably compact if and only if it is a retract of a spectral locale (i.e. the frame of ideals of a distributive lattice). Proof. A stably compact locale X is a retract of the frame of ideals Idl(X). Since every idempotent splits in the category of locales: Corollary The category of stably compact locales is equivalent to the splitting of idempotents Split(Spec) of the category Spec of spectral locales.
◮ An object of Split(Spec) is an idempotent (i.e. f : X → X s.t. f ◦ f = f ) in Spec. ◮ A morphism g: (f : X → X) → (f ′ : X′ → X′) in Split(Spec) is a
continuous map g: X → X′ in Spec such that f ′ ◦ g = g = g ◦ f .
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Spectral locales
A relation r ⊆ D × D′ between distributive lattices D and D′ is approximable if
- 1. ra
def
= {b ∈ D′ | a r b} is a filter for each a ∈ D,
- 2. r−b
def
= {a ∈ D | a r b} is an ideal of D for each b ∈ D′,
- 3. a r 0′ =
⇒ a = 0,
- 4. a r b ∨′ c =
⇒ (∃b′, c′ ∈ D) a ≤ b′ ∨ c′ & b′ r b & c′ r c.
Distributive lattices and approximable relations form a category DLAP with identities ≤D and relational compositions. Proposition The category DLAP is equivalent to the category of spectral locales.
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Strong proximity lattices (Jung & Sünderhauf)
A strong proximity lattice is an object of Split(DLAP), i.e. a distributive lattice D equipped with an idempotent relation ≺ such that
- 1. ↓ a
def
= {b ∈ D | b ≺ a} is an ideal
- 2. ↑ a
def
= {b ∈ D | b ≻ a} is a filter
- 3. a ≺ 0 =
⇒ a = 0
- 4. a ≺ b ∨ c =
⇒ (∃b′ ≺ b) (∃c′ ≺ c) a ≤ b′ ∨ c′
- 5. 1 ≺ a =
⇒ a = 1
- 6. a ∧ b ≺ c =
⇒ (∃a′ ≻ a) (∃b′ ≻ b) a′ ∧ b′ ≤ c.
Morphisms of strong proximity lattices are approximable relations. Remark A strong proximity lattice (D, ≺) represents a stably compact locale X such that
Ω(X) ∼ = Rounded ideals of (D, ≺).
An ideal I ⊆ D is rounded if a ∈ I ⇐
⇒ (∃b ≻ a) b ∈ I.
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Continuous entailment relations (Coquand & Zhang)
An entailment relation on a set S is a binary relation ⊢ on the finite subsets of S such that
a ∈ S a ⊢ a A ⊢ B A′, A ⊢ B, B′ A ⊢ B, a a, A ⊢ B A ⊢ B
where “,” denotes a union. Remark An entailment relation (S, ⊢) presents a distributive lattice generated with generators S and relations A ≤ B for A ⊢ B. An entailment relation (S, ⊢) is continuous if it is equipped with an idempotent relation ≺ on S such that
(∃C) A ≺U C ⊢ B ⇐ ⇒ (∃D) A ⊢ D ≺L B
where
A ≺U B
def
⇐ ⇒ (∀b ∈ B) (∃a ∈ A) a ≺ b A ≺L B
def
⇐ ⇒ (∀a ∈ A) (∃b ∈ B) a ≺ b.
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Continuous entailment relations
Proposition The category of continuous entailment relations is equivalent to that of strong proximity lattices. Proof.
◮ If (D, ≺) is a strong proximity lattice, then (D, ⊢D) defined by A ⊢D B
def
⇐ ⇒
- A ≤D
- B
together with ≺ is a continuous entailment relation.
◮ If (S, ⊢, ≺) is a continuous entailment relation, then the lattice DS
generated by (S, ⊢) together with the relation ≪ on DS defined by
- i<N
- Ai ≪
- j<M
- Bj
def
⇐ ⇒ ∀i < N∀j < M∃C [Ai ≺U C ⊢ Bj]
is a strong proximity lattice.
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Generated continuous entailment relations
Let R be a set of pairs of finite subsets of a set S (called R a set of axioms). An entailment relation (S, ⊢) is generated by R if it is the smallest entailment relation on S that contains R, i.e. ⊢ is generated by the following rules:
(A, B) ∈ R A ⊢ B a ∈ S a ⊢ a A ⊢ B A′, A ⊢ B, B′ A ⊢ B, a a, A ⊢ B A ⊢ B
Lemma Let (S, ⊢) be the entailment relation generated by a set R of axioms. Then the dual ⊣ is generated by Rop def
= {(B, A) | (A, B) ∈ R}.
Proposition Let (S, ⊢) be the entailment relation generated by a set R of axioms, and let ≺ be an idempotent relation on S. Then (S, ⊢, ≺) is a continuous entailment if and only if for each A and B
- 1. A ≺U C & (C, D) ∈ R =
⇒ (∃D′) A ⊢ D′ ≺L D
- 2. (C, D) ∈ R & D ≺L B =
⇒ (∃C′) C ≺U C′ ⊢ B
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Continuous entailment relation as the space its models
Definition A model of a continuous entailment relation (S, ⊢, ≺) is a subset
α ⊆ S such that
- 1. A ⊆ α =
⇒ (∃b ∈ B) b ∈ α for each A ⊢ B,
- 2. a ∈ α =
⇒ (∃b ≺ a) b ∈ α.
Example If X is a locale presented by a strong proximity lattice (D, ≺), the Scott topology Σ(X) can be defined as the space of its rounded ideals.
Σ(X) can be presented by an entailment relation on ⊠D
def
= {⊠a | a ∈ D} generated by ∅ ⊢ ⊠0 ⊠ a, ⊠b ⊢ ⊠(a ∨ b) ⊠ a ⊢ ⊠b (a ≥ b)
together with the idempotent relation ⊠a ≺⊠ ⊠b
def
⇐ ⇒ a ≻ b.
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Plan
- 1. Strong proximity lattices
- 2. De Groot duality
- 3. Applications
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Duality in strong proximity lattices
Definition
◮ The dual Dd of a strong proximity lattice (D, ≺) is
- Dd, ≻
- ,
where Dd is the dual lattice of D.
◮ The dual Sd of a continuous entailment relation (S, ⊢, ≺) is (S, ⊣, ≻).
Proposition The equivalence between continuous entailment relations and strong proximity relations commutes with the dualities. Question If X is the stably compact locale presented by (D, ≺), does (Dd, ≻) present the de Groot dual of X?
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De Groot duality
The de Groot dual X△ of a stably compact locale X is the frame of Scott open filters on Ω(X). Definition Let X be a stably compact locale. The upper powerlocale PU(X) is a locale whose points (i.e. models) are Scott open filters on Ω(X). The de Groot dual of X can be characterized by PU(X) ∼
= Σ(Xd).
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De Groot duality in strong proximity lattices
Proposition Given a strong proximity lattice (D, ≺), let X and Y be stably compact locales presented by (D, ≺) and (Dd, ≻) respectively. Then
PU(X) ∼ = Σ(Y).
- Proof. The upper powerlocale PU(X) is presented by an entailment
relation on ✷D
def
= {✷a | a ∈ D} generated by ∅ ⊢ ✷1 ✷a, ✷b ⊢ ✷(a ∧ b) ✷a ⊢ ✷b (a ≤ b)
together with the idempotent relation ✷a ≺✷ ✷b
def
⇐ ⇒ a ≺ b.
The Scott topology Σ(Y) is presented by an entailment relation on
⊠D
def
= {⊠a | a ∈ D} generated by ∅ ⊢ ⊠1 ⊠ a, ⊠b ⊢ ⊠(a ∧ b) ⊠ a ⊢ ⊠b (a ≤ b)
together with the idempotent relation ⊠a ≺⊠ ⊠b
def
⇐ ⇒ a ≺ b.
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Plan
- 1. Strong proximity lattices
- 2. De Groot duality
- 3. Applications
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The lower and upper powerlocales
Let X be a stably compact locale presented by a strong proximity lattice (D, ≺).
◮ The lower powerlocale PL(X) is presented by an entailment
relation on ✸D
def
= {✸a | a ∈ D} generated by ✸0 ⊢ ∅ ✸(a ∨ b) ⊢ ✸a, ✸b ✸a ⊢ ✸b (a ≤ b)
together with the idempotent relation ✸a ≺✸ ✸b
def
⇐ ⇒ a ≺ b. ◮ The upper powerlocale PU(X) is presented by an entailment
relation on ✷D
def
= {✷a | a ∈ D} generated by ∅ ⊢ ✷1 ✷a, ✷b ⊢ ✷(a ∧ b) ✷a ⊢ ✷b (a ≤ b)
together with the idempotent relation ✷a ≺✷ ✷b
def
⇐ ⇒ a ≺ b.
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The lower and upper powerlocales
Proposition If X is stably compact then PL(X)d ∼
= PU(Xd) and PU(X)d ∼ = PL(Xd).
Proof. Suppose X is presented by a strong proximity lattice (D, ≺).
◮ PL(X) is presented by an entailment relation generated by ✸0 ⊢ ∅ ✸(a ∨ b) ⊢ ✸a, ✸b ✸a ⊢ ✸b (a ≤ b)
with the idempotent relation ✸a ≺✸ ✸b
def
⇐ ⇒ a ≺ b. ◮ PL(X)d is presented by an entailment relation generated by ∅ ⊢ ✸0 ✸a, ✸b ⊢ ✸(a ∨ b) ✸b ⊢ ✸a (a ≤ b)
with the idempotent relation ✸a ≺✸ ✸b
def
⇐ ⇒ a ≻ b. ◮ This is just a presentation of PU(Xd).
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The Vietoris powerlocales
Let X be a stably compact locale presented by a strong proximity lattice (D, ≻). The Vietoris powerlocale PV(X) is presented by an entailment relation on ✸D ∪ ✷D generated by
✸0 ⊢ ∅ ✸(a ∨ b) ⊢ ✸a, ✸b ✸a ⊢ ✸b (a ≤ b) ∅ ⊢ ✷1 ✷a, ✷b ⊢ ✷(a ∧ b) ✷a ⊢ ✷b (a ≤ b) ✷a, ✸b ⊢ ✸(a ∧ b) ✷(a ∨ b) ⊢ ✷a, ✸b
The idempotent relation associated with PV(X) is ≺✸ ∪ ≺✷. Proposition If X is stably compact then PV(X)d ∼
= PV(Xd).
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The space of valuations
A probability valuation on a locale X is a Scott continuous map
µ: Ω(X) → − − → [0, 1] to the lower reals − − → [0, 1] satisfying µ(0) = 0, µ(1) = 1, and the modular law: µ(x) + µ(y) = µ(x ∧ y) + µ(x ∨ y).
Let V(X) be the locale whose points are valuations on X. A covaluation on X is a Scott continuous map ν : Ω(X) → ←
− − [0, 1]
to the upper reals ←
− − [0, 1] satisfying ν(1) = 0, ν(0) = 1, and the
modular law. Let C(X) be a locale whose points are covaluations on X.
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The space of valuations
Proposition If X is a locale presented by a strong proximity lattice (D, ≺), then the space of valuations V(X) is presented by an entailment relation on
{p, a | p ∈ Q & a ∈ D} generated by the axioms ∅ ⊢ p, 0 (p < 0) p, 0 ⊢ ∅ (0 < p) ∅ ⊢ p, 1 (p < 1) p, 1 ⊢ ∅ (1 < p) p, a ⊢ q, b (p ≥ q & a ≤ b) p, a, q, b ⊣⊢ r, a ∧ b, s, a ∨ b (p + q = r + s)
with an idempotent relation p, a ≺V q, b
def
⇐ ⇒ p > q & a ≺ b.
- Note. Each generator p, a expresses p < µ(a).
Proposition If X is a stably compact locale, then V(X)d ∼
= C(Xd).
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