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Duality for sheaves of distributive-lattice-ordered algebras over - - PowerPoint PPT Presentation

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Duality for sheaves of distributive-lattice-ordered algebras over stably compact spaces Sam van Gool (joint work with Mai Gehrke) LIAFA,

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case

Duality for sheaves of distributive-lattice-ordered algebras over stably compact spaces

Sam van Gool (joint work with Mai Gehrke)

LIAFA, Université Paris Diderot (FR) & Radboud Universiteit Nijmegen (NL)

6 August 2013 BLAST Chapman University, Orange, CA

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case

This talk in a picture

F(Y ) = D E Y p a (D/y)∗ q Y ∂ X = D∗ X

  • a

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Sheaves and étale spaces

Definition of étale space

Let V be a variety of abstract algebras, (Y , ρ) a topological space. Let (Ay)y∈Y be a Y -indexed family of V-algebras. Let E :=

y∈Y Ay, with p : E ։ Y the natural surjection.

Suppose τ is a topology on E such that p : (E, τ) ։ (Y , ρ) is a local homeomorphism: any point has an open neighbourhood on which p has a right inverse. p : (E, τ) ։ (Y , ρ) is called an étale space of V-algebras.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Sheaves and étale spaces

Sheaf from an étale space

Let p : (E, τ) ։ (Y , ρ) be an étale space of V-algebras. For any U ∈ ρ, write F(U) for the set of local sections over U: F(U) := {s : U → E continuous s.t. p ◦ s = idU}. Note: F(U) is a V-algebra (being a subalgebra of

y∈U Ay).

If U ⊆ V , there is a natural restriction map F(V ) → F(U). F is called the sheaf associated with p.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Sheaves and étale spaces

Definition of sheaf

In general, a sheaf F on Y consists of the data:

For each open U, a V-algebra F(U) (“local sections”); For each open U ⊆ V , a V-homomorphism ()|U : F(V ) → F(U) (“restriction maps”);

such that the appropriate diagrams commute, satisfying the following patching property:

For any open cover (Ui)i∈I of an open set U, (si)i∈I a “compatible family” of local sections, i.e., si|Ui ∩Uj = sj|Ui ∩Uj for all i, j ∈ I. there exists a unique s ∈ F(U) such that s|Ui = si for all i ∈ I.

F(Y ) is called the algebra of global sections of the sheaf F.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Sheaves and étale spaces

Sheaves vs. étale spaces

Fact Any sheaf arises from an étale space, and vice versa.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product representation

Let A be an abstract algebra. A Boolean product representation of A is a sheaf F on a Boolean space Y such that A is isomorphic to the algebra of global sections of F. Equivalent: a subdirect embedding A ֌

y∈Y Ay satisfying:

(Open equalizers) For any a, b ∈ A, the equalizer a = b := {y ∈ Y | ay = by} is open; (Patch) For K clopen in Y , a, b ∈ A, there exists c ∈ A such that a|K = c|K and b|K c = c|K c.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product, pictorially

F Y

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product, pictorially

F Y y

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product, pictorially

A/y F Y y

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product, pictorially

A/y F Y y y′

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product, pictorially

A/y A/y′ F Y y y′

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product, pictorially

A/y A/y′ F Y y y′

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product, pictorially

A/y A/y′ F Y K y y′

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product, pictorially

A/y A/y′ F Y K y y′ a

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product, pictorially

A/y A/y′ F Y K K c y y′ a b

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean product, pictorially

A/y A/y′ F Y K K c y y′ a b c

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Lattices of congruences

Theorem (Comer 1971, Burris & Werner 1980) Boolean product representations of A are in a natural one-to-one correspondence with relatively complemented distributive lattices of permuting congruences on A.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Boolean sum decompositions

Let D be a distributive lattice. Theorem (Gehrke 1991) Boolean product representations D ֌

y∈Y Dy are in a natural

  • ne-to-one correspondence with Boolean sum decompositions of

the Stone dual space X of D into the Stone dual spaces (Xy)y∈Y

  • f the lattices (Dy)y∈Y .

Also see [Hansoul & Vrancken-Mawet 1984] for a version for the Priestley dual spaces.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Dual characterization, pictorially

F(Y ) = D E Y p

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Dual characterization, pictorially

F(Y ) = D E Y p Y X = D∗ X

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Dual characterization, pictorially

A/y F(Y ) = D E Y p y Y X = D∗ X

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Dual characterization, pictorially

A/y F(Y ) = D E Y p y (D/y)∗ Y X = D∗ X y

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Dual characterization, pictorially

A/y F(Y ) = D E Y p y (D/y)∗ Y X = D∗ X y

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Dual characterization, pictorially

A/y F(Y ) = D E Y p y (D/y)∗ Y X = D∗ X y

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Dual characterization, pictorially

A/y F(Y ) = D E Y p y (D/y)∗ q Y X = D∗ X y

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Dual characterization, pictorially

A/y F(Y ) = D E Y p y a (D/y)∗ q Y X = D∗ X y

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Dual characterization, pictorially

A/y F(Y ) = D E Y p y a (D/y)∗ q Y X = D∗ X y

  • a

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

This talk in a picture

F(Y ) = D E Y p a (D/y)∗ q Y ∂ X = D∗ X

  • a

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Boolean products

Question

What if Y is no longer a Boolean space?

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact spaces

Motivation

Many interesting sheaf representations use a base space which is spectral or compact Hausdorff. Stably compact spaces form a common generalization of these two classes.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact spaces

Stably compact spaces

“Generalisation of compact Hausdorff to T0-setting” Definition Stably compact space = T0, Sober, Locally compact, Intersection of compact saturated is compact.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact spaces

De Groot dual and patch topology

For any topological space (Y , ρ), define its de Groot dual ρ∂ := U ⊆ Y | Y \ U is compact saturated in ρtop

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De Groot dual and patch topology

For any topological space (Y , ρ), define its de Groot dual ρ∂ := U ⊆ Y | Y \ U is compact saturated in ρtop Fact: If (Y , ρ) is stably compact, then so is Y ∂ := (Y , ρ∂).

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact spaces

De Groot dual and patch topology

For any topological space (Y , ρ), define its de Groot dual ρ∂ := U ⊆ Y | Y \ U is compact saturated in ρtop Fact: If (Y , ρ) is stably compact, then so is Y ∂ := (Y , ρ∂). Define ρp := ρ ∨ ρ∂, the patch topology.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact spaces

De Groot dual and patch topology

For any topological space (Y , ρ), define its de Groot dual ρ∂ := U ⊆ Y | Y \ U is compact saturated in ρtop Fact: If (Y , ρ) is stably compact, then so is Y ∂ := (Y , ρ∂). Define ρp := ρ ∨ ρ∂, the patch topology. Fact: (Y , ρp) is a compact Hausdorff space.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact spaces

De Groot dual and patch topology

For any topological space (Y , ρ), define its de Groot dual ρ∂ := U ⊆ Y | Y \ U is compact saturated in ρtop Fact: If (Y , ρ) is stably compact, then so is Y ∂ := (Y , ρ∂). Define ρp := ρ ∨ ρ∂, the patch topology. Fact: (Y , ρp) is a compact Hausdorff space. Let y ≤ y′ ⇐ ⇒ y′ ∈ {y}, the specialization order of ρ.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact spaces

De Groot dual and patch topology

For any topological space (Y , ρ), define its de Groot dual ρ∂ := U ⊆ Y | Y \ U is compact saturated in ρtop Fact: If (Y , ρ) is stably compact, then so is Y ∂ := (Y , ρ∂). Define ρp := ρ ∨ ρ∂, the patch topology. Fact: (Y , ρp) is a compact Hausdorff space. Let y ≤ y′ ⇐ ⇒ y′ ∈ {y}, the specialization order of ρ. Fact: ≤ is a closed subspace of (Y × Y , ρp × ρp).

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact spaces

De Groot dual and patch topology

For any topological space (Y , ρ), define its de Groot dual ρ∂ := U ⊆ Y | Y \ U is compact saturated in ρtop Fact: If (Y , ρ) is stably compact, then so is Y ∂ := (Y , ρ∂). Define ρp := ρ ∨ ρ∂, the patch topology. Fact: (Y , ρp) is a compact Hausdorff space. Let y ≤ y′ ⇐ ⇒ y′ ∈ {y}, the specialization order of ρ. Fact: ≤ is a closed subspace of (Y × Y , ρp × ρp). So (Y , ρp, ≤) is a compact ordered space (Nachbin 1965).

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact spaces

Compact ordered spaces

Conversely, given a compact ordered space (Y , π, ≤), let π↓ the topology of open down-sets. Then (Y , π↓) is a stably compact space, and (π↓)∂ = π↑.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact spaces

Compact ordered spaces

Conversely, given a compact ordered space (Y , π, ≤), let π↓ the topology of open down-sets. Then (Y , π↓) is a stably compact space, and (π↓)∂ = π↑. Fact The categories of stably compact spaces and compact ordered spaces are isomorphic.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Proximity lattices

Representing stably compact spaces

Example (Open basis presentation) X stably compact space D lattice-basis of open sets for X Define U R V iff there exists compact saturated K ⊆ X such that U ⊆ K ⊆ V Fact: X can be recovered as the space of “round prime ideals”

  • f R.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Proximity lattices

Spectral spaces with retractions

Fact (Johnstone, 1982) A topological space X is stably compact iff X X

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Spectral spaces with retractions

Fact (Johnstone, 1982) A topological space X is stably compact iff there exists a spectral space Y Y Y X X

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Proximity lattices

Spectral spaces with retractions

Fact (Johnstone, 1982) A topological space X is stably compact iff there exists a spectral space Y and a continuous retraction of Y onto X. Y Y Y X X f

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Proximity lattices

Duality for spectral spaces with continuous maps

Fact: DLj ∼ =op SpecSpc Here, SpecSpc: spectral spaces with continuous maps, and DLj: distributive lattices with j-morphisms: Definition A relation H ⊆ D × E between distributive lattices D and E is called a j-morphism iff:

≥ ◦ H ◦ ≥ = H a H B ⇐ ⇒ ∀b ∈ B aHb A H b ⇐ ⇒ ∀a ∈ A aHb If A H b then ∃B ⊆ω H[A] such that b ≤ B.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Proximity lattices

Duality for stably compact spaces

Definition A join-strong proximity lattice is a pair (D, R) where D is a distributive lattice, R−1 : D → D is a j-morphism, and R ◦ R = R.

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Duality for stably compact spaces

Definition A join-strong proximity lattice is a pair (D, R) where D is a distributive lattice, R−1 : D → D is a j-morphism, and R ◦ R = R. Fact The categories of stably compact spaces and join-strong proximity lattices are equivalent.

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Duality for stably compact spaces

Definition A join-strong proximity lattice is a pair (D, R) where D is a distributive lattice, R−1 : D → D is a j-morphism, and R ◦ R = R. Fact The categories of stably compact spaces and join-strong proximity lattices are equivalent. Proof. Stably compact spaces are retracts of spectral spaces.

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Duality for stably compact spaces

Definition A join-strong proximity lattice is a pair (D, R) where D is a distributive lattice, R−1 : D → D is a j-morphism, and R ◦ R = R. Fact The categories of stably compact spaces and join-strong proximity lattices are equivalent. Proof. Stably compact spaces are retracts of spectral spaces. Therefore, duals of stably compact spaces are retracts of distributive lattices in the category DLj.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact sum decompositions

The case of MV-algebras

Theorem (Gehrke, Marra, vG 2012) The Priestley dual space X of the distributive lattice underlying an MV-algebra A decomposes as a stably compact sum over the base space Y of prime MV ideals of A.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact sum decompositions

Stably compact sum decompositions

Definition A stably compact sum decomposition of a Priestley space X is a continuous surjection q : X ։ Y ∂, with Y stably compact, satisfying the following dual patching property: (P) Let (Ui)n

i=1 be any finite cover of Y by ρ∂-open sets, and let

( ai)n

i=1 be any finite collection of clopen downsets of X such

that

  • ai ∩ q−1(Ui ∩ Uj) =

aj ∩ q−1(Ui ∩ Uj) holds for any i, j ∈ {1, . . . , n}. Then the set n

i=1(

ai ∩ q−1(Ui)) is a clopen downset in X.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Stably compact sum decompositions

Property (P), pictorially

X q Y ∂

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Property (P), pictorially

X q Y ∂ U1

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Property (P), pictorially

X q Y ∂

  • a1

U1

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Property (P), pictorially

X q Y ∂

  • a2
  • a1

U1 U2

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Property (P), pictorially

X q Y ∂

  • a2
  • a1

U1 U2 y Xy

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Property (P), pictorially

X q Y ∂

  • a2
  • a1

U1 U2 y Xy n

i=1(

ai ∩ q−1(Ui))

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Spectral sum yields sheaf

Theorem (Gehrke, Marra, vG 2012) If X is the Priestley space of a distributive lattice A, then any stably compact sum decomposition q : X ։ Y ∂ yields a sheaf representation of A over Y . Example For an MV-algebra A, there are two natural stably compact sum decompositions of the dual space X, each of which yields a sheaf representation of A: one over its prime, the other over its maximal spectrum.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Fitted sheaves

Fitted sheaves

Question: which sheaves can be captured by such a decomposition?

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Fitted sheaves

Question: which sheaves can be captured by such a decomposition? Let B a basis for the base space Y . Call a sheaf F fitted for B if, for each U ∈ B, the restriction map F(Y ) → F(U) is surjective. (“Fitted for O(Y )” = “flabby” or “flasque”...)

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Lattices of congruences, revisited

Let F be a sheaf of distributive lattices on a topological space Y which is fitted for a lattice basis B for Y with A := F(Y ). For U ∈ B, define θF(U) := ker(F(Y ) ։ F(U)). Proposition The function θF : Bop → ConDL(A) is a lattice homomorphism, and any two congruences in the image of θF permute.

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Sheaf yields decomposition map

Given a sheaf F fitted for B, lift this lattice homomorphism θF : Bop → ConDL(A) to θF : O(Y ∂) → ConDL(A). Note that ConDL(A) ∼ = O(X), where X is the Priestley dual space of the distributive lattice A. By pointless duality, we obtain a continuous map q : X → Y ∂.

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Lifting to frames

Lemma (Lifting) Suppose that B is a lattice basis for the open sets of a stably compact space Y and that h : Bop → F is a lattice homomorphism from Bop into a frame F. Then the function h : O(Y ∂) → F defined by h(W ) :=

  • {h(U) | U ∈ B, Uc ⊆ W }

is a frame homomorphism. Proof based on strong proximity lattice of (O, K)-pairs by Jung & Sünderhauf (1996).

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Proof of lifting lemma

To show: h(W ) := {h(U) | U ∈ B, Uc ⊆ W } preserves .

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Proof of lifting lemma

To show: h(W ) := {h(U) | U ∈ B, Uc ⊆ W } preserves . Enough: h(

i∈I Wi) ≤ i∈I h(Wi).

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Proof of lifting lemma

To show: h(W ) := {h(U) | U ∈ B, Uc ⊆ W } preserves . Enough: h(

i∈I Wi) ≤ i∈I h(Wi).

From the fact that B is a basis, deduce that Wi = {V ∈ O(Y ∂) | ∃U ∈ B : V ⊆ Uc ⊆ Wi}.

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Proof of lifting lemma

To show: h(W ) := {h(U) | U ∈ B, Uc ⊆ W } preserves . Enough: h(

i∈I Wi) ≤ i∈I h(Wi).

From the fact that B is a basis, deduce that Wi = {V ∈ O(Y ∂) | ∃U ∈ B : V ⊆ Uc ⊆ Wi}. So, if U ∈ B and Uc ⊆

i∈I Wi, by compactness pick finite

cover F ⊆ {V ∈ O(Y ∂) | ∃i ∈ I, U ∈ B : V ⊆ Uc ⊆ Wi}.

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Proof of lifting lemma

To show: h(W ) := {h(U) | U ∈ B, Uc ⊆ W } preserves . Enough: h(

i∈I Wi) ≤ i∈I h(Wi).

From the fact that B is a basis, deduce that Wi = {V ∈ O(Y ∂) | ∃U ∈ B : V ⊆ Uc ⊆ Wi}. So, if U ∈ B and Uc ⊆

i∈I Wi, by compactness pick finite

cover F ⊆ {V ∈ O(Y ∂) | ∃i ∈ I, U ∈ B : V ⊆ Uc ⊆ Wi}. For each V ∈ F, pick UV ∈ B, iV ∈ I, with V ⊆ (UV )c ⊆ WiV and Uc ⊆ F.

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Proof of lifting lemma

To show: h(W ) := {h(U) | U ∈ B, Uc ⊆ W } preserves . Enough: h(

i∈I Wi) ≤ i∈I h(Wi).

From the fact that B is a basis, deduce that Wi = {V ∈ O(Y ∂) | ∃U ∈ B : V ⊆ Uc ⊆ Wi}. So, if U ∈ B and Uc ⊆

i∈I Wi, by compactness pick finite

cover F ⊆ {V ∈ O(Y ∂) | ∃i ∈ I, U ∈ B : V ⊆ Uc ⊆ Wi}. For each V ∈ F, pick UV ∈ B, iV ∈ I, with V ⊆ (UV )c ⊆ WiV and Uc ⊆ F. Then h(U) ≤ h(

V ∈F UV ) ≤ V ∈F h(UV ) ≤ i∈I h(Wi).

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The decomposition map

Let θF : O(Y ∂) → ConDL(A) be the frame homomorphism associated to a sheaf F.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Fitted sheaves

The decomposition map

Let θF : O(Y ∂) → ConDL(A) be the frame homomorphism associated to a sheaf F. The function p : (X, τ p) → (Y , ρ∂) dual to θF is given by: p(x) = max{y ∈ Y | x ∈ Xy}.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Fitted sheaves

The decomposition map

Let θF : O(Y ∂) → ConDL(A) be the frame homomorphism associated to a sheaf F. The function p : (X, τ p) → (Y , ρ∂) dual to θF is given by: p(x) = max{y ∈ Y | x ∈ Xy}. This shows that an analogue of the function k from MV-algebras1 is available in the context of any fitted sheaf representation!

1See Mai Gehrke’s talk yesterday afternoon.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Conclusions

Main theorem

Theorem (Gehrke, vG 2013) A fitted sheaf representation of a distributive lattice A over a stably compact space Y yields a stably compact sum decomposition of the Priestley dual space X of A over Y ∂.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Conclusions

This talk in a picture

F(Y ) = D E Y p a (D/y)∗ q Y ∂ X = D∗ X

  • a

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Conclusions

Analogy with Boolean case

Sheaf over Boolean space Sheaf over stably compact space

  • Rel. comp. distributive lat-

tice of congruences Strong proximity lattice of congruences Boolean sum decomposition Stably compact sum decomposition

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Conclusions

Further work

To retrieve the topology of the dual space from the topologies

  • n the subspaces and on the base space;

To apply these results to more general and to other classes of lattice-ordered algebras.

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Duality for sheaves - Boolean case Stably compact spaces Duality for sheaves - stably compact case Conclusions

Duality for sheaves of distributive-lattice-ordered algebras over stably compact spaces

Sam van Gool (joint work with Mai Gehrke)

LIAFA, Université Paris Diderot (FR) & Radboud Universiteit Nijmegen (NL)

6 August 2013 BLAST Chapman University, Orange, CA

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