Pointfree convergence Jean Goubault-Larrecq and Fr eric Mynard ed - - PowerPoint PPT Presentation

Pointfree Convergence

Pointfree convergence

Jean Goubault-Larrecq∗ and Fr´ ed´ eric Mynard∗∗

∗ LSV, ENS Cachan, CNRS, Universit´

e Paris-Saclay

∗∗ New Jersey City University

Dagstuhl ⊣ Dualities

Pointfree Convergence

Outline

1 Introduction 2 The convergence space/focale duality 3 Sobrification 4 Relation to Stone duality 5 Conclusion

Pointfree Convergence Introduction

Outline

1 Introduction 2 The convergence space/focale duality 3 Sobrification 4 Relation to Stone duality 5 Conclusion

Pointfree Convergence Introduction

Stone Duality

Theorem There is an adjunction Top : OStone ⊣ ptStone : Loc = Frmop: OStone maps each topological space X to its lattice of open subsets ptStone maps each frame L to space of completely prime filters Loc side: pointfree topology This talk: an extension to convergence spaces . . . leading to pointfree convergence

Pointfree Convergence Introduction

Convergence spaces

Let FL be the set of filters on a (bounded) lattice L FPX is the set of filters of subsets of X For x ∈ X, ˙ x = {A ⊆ X | x ∈ A} is the principal filter at x Convergence spaces (X, →) where →⊆ FPX × X such that: (Triv.) ˙ x → x (Mono.) If F → x and F ⊆ G then G → x.

Pointfree Convergence Introduction

Reformulating convergence spaces

Given (X, →), let lim F = {x ∈ X | F → x}. Convergence spaces (X, lim) where lim: FPX → PX such that: (Triv.) x ∈ lim ˙ x (Mono.) lim is monotonic. Almost lends itself to a pointfree formulation: Replace PX by an abstract lattice L Axiom (Triv.) seems to cause a problem (requires points). . .

Pointfree Convergence The convergence space/focale duality

Outline

1 Introduction 2 The convergence space/focale duality 3 Sobrification 4 Relation to Stone duality 5 Conclusion

Pointfree Convergence The convergence space/focale duality

Convergence lattices and focales

Definition A convergence lattice is a lattice L together with a monotonic map limL : FL → L. So we get (Mono.). . . and ignore (Triv.) altogether They form a category CL. Morphisms imitate continuity for f , translated to f −1: ϕ: L → L′ morphism ⇔ lattice homomorphism + for every F ∈ FL′, limL′ F ≤ ϕ(limL ϕ−1(F)). Defines a functor L: Conv → Foc = CLop: L(X, →) = (PX, lim), L(f ) = f −1.

Pointfree Convergence The convergence space/focale duality

Retrieving points

Definition (Point) A point in a convergence lattice L is a prime filter x ∈ FL such that limL x ∈ x. This is (Triv.): ˙ x is an ultrafilter (=a prime filter) such that lim ˙ x ∈ ˙ x (⇔ x ∈ lim ˙ x) I.e., we represent points x in X by (certain) compact ultrafilters ˙ x. For an ultrafilter of subsets U, lim U ∈ U iff U is compact.

Pointfree Convergence The convergence space/focale duality

Retrieving points. . . and convergence

Fix a convergence lattice L. Definition Let pt L be its set of points (=compact prime filters), with convergence given by F → x iff x ∈ (limL F♭)♯. For ℓ ∈ L, ℓ♯ ∈ P pt Lˆ ={x ∈ pt L | ℓ ∈ x} “x ∈ ℓ♯ ⇔ ℓ ∈ x”

♭ is Kowalsky sum: for F ∈ FP pt L,

F♭ ∈ FL = {ℓ ∈ L | ℓ♯ ∈ F} FP pt L

♭

• lim P pt L

FL

limL

L

♯

Pointfree Convergence The convergence space/focale duality

Retrieving points. . . and convergence

Fix a convergence lattice L. Definition Let pt L be its set of points, with F → x iff x ∈ (limL F♭)♯. Defines a functor pt: Foc = CLop → Conv On morphisms ϕ: L′ → L in CL, pt ϕ = ϕ−1. Theorem Conv : L ⊣ pt : Foc. Unit ηX : X → pt L X: ηX(x) = ˙ x Counit ǫL : L → L pt L: ǫL(ℓ) = ℓ♯.

Pointfree Convergence Sobrification

Outline

1 Introduction 2 The convergence space/focale duality 3 Sobrification 4 Relation to Stone duality 5 Conclusion

Pointfree Convergence Sobrification

Sobrification and sober spaces

In Top (not Conv), we have: an adjunction Top : OStone ⊣ ptStone : Loc = Frmop induces a monad S = ptStone OStone on Top (sobrification) S is an idempotent monad: A space X is sober iff X ∼ = S(Y ) for some Y iff X ∼ = S(X) We shall study the corresponding monad pt L on Conv. Beware: pt L is not idempotent.

Pointfree Convergence Sobrification

Quasi-sober convergence spaces

Let X be a convergence space. Definition A generic point of a filter F ∈ FPX is a point x ∈ X such that lim F = lim ˙ x. Usual notion of a generic point when X is a topological space. Definition X is quasi-sober iff every compact ultrafilter has a generic point. Remember: U compact ⇔ lim U ∈ U. Note: When X is a topological space, X is quasi-sober iff every irreducible closed subset has a generic point (= “sober minus T0”).

Pointfree Convergence Sobrification

pt L is quasi-sober

Proposition For every convergence lattice L, X = pt L is quasi-sober.

• Proof. for every compact ultrafilter U, U♭ is a generic point of

U. This alone does not characterize the convergence spaces of the form pt L. Let ϕ: pt L X → X be a fixed map, the designated limit map Think of it as

♭ when X = pt L.

We shall impose further conditions on ϕ. The most important notion is that of a tile (wrt. ϕ)

Pointfree Convergence Sobrification

Tiles

Fix a map ϕ: pt L X → X (ϕ =

♭ if X = pt L)

Definition A tile is a subset S of X such that for every U ∈ pt L X, S ∈ U iff ϕ(U) ∈ S. Tiles are an attempt at characterizing the subsets ℓ♯ without referring to L: Proposition In X = pt L, every subset of the form ℓ♯ ⊆ X is a tile. Apart from that, still a mysterious notion to us. Examples: In a sober topological space, every open is a tile In a Hausdorff convergence space, every subset is a tile.

Pointfree Convergence Sobrification

The poset of tiles

Proposition The poset TX of all tiles is a Boolean subalgebra of PX. In particular, in a sober topological space, every finite union of crescents is a tile. Take L = TX: can we make it a convergence lattice? Need to find limL: possible under some additional conditions: repleteness, tiledness, and separation.

Pointfree Convergence Sobrification

Replete, tiled, and separated convergence spaces

Definition X is replete iff for every filter of subsets F, lim F is a tile. True when X = pt L: lim F = (limL F♭)♯ is a tile.

Pointfree Convergence Sobrification

Replete, tiled, and separated convergence spaces

Definition X is replete iff for every filter of subsets F, lim F is a tile. True when X = pt L: lim F = (limL F♭)♯ is a tile. Definition X is tiled iff convergence only depends on the tiles: if F ∩ TX = G ∩ TX, then lim F = lim G. True when X = pt L, because F♭ only depends on the tiles ℓ♯ in F.

Pointfree Convergence Sobrification

Replete, tiled, and separated convergence spaces

Definition X is replete iff for every filter of subsets F, lim F is a tile. True when X = pt L: lim F = (limL F♭)♯ is a tile. Definition X is tiled iff convergence only depends on the tiles: if F ∩ TX = G ∩ TX, then lim F = lim G. True when X = pt L, because F♭ only depends on the tiles ℓ♯ in F. Definition X is separated iff for all x = y, there is a tile that contains one but not the other. True when X = pt L; remember: ℓ ∈ x iff x ∈ ℓ♯.

Pointfree Convergence Sobrification

Temperance

. . . is a strong form of sobriety. Definition A convergence space X is temperate iff there is a map ϕ: pt L X → X such that: for every U ∈ pt L X, U → ϕ(U) X is replete, tiled, and separated wrt. ϕ. Note: for a T0 topological space, sober=quasi-sober=temperate. Every convergence space of the form pt L is temperate.

Pointfree Convergence Sobrification

The range of pt

Theorem (Range of pt=temperate spaces) X is isomorphic to a convergence space of the form pt L iff X is temperate. In that case, we can take TX for L.

• Proof. Let L = TX.

For F ∈ FPX, define limL(F ∩ L) = lim F. Makes sense since X is tiled. Yields a tile since X is replete, so (L, limL) convergence lattice. Define γX : X → pt L by γX(x) = ˙ x ∩ L. γX surjective by Zorn’s Lemma + prime=maximal for filters on a Boolean algebra (L = TX) γX injective since X is separated Finally, F → x iff γX[F] → γX(x): verification.

Pointfree Convergence Sobrification

Tee-totalers?

An even stronger form of sobriety would hold of convergence spaces X such that X ∼ = pt L X. Theorem The following are equivalent: ηX : X → pt L X is an isomorphism ηX : X → pt L X is onto every compact ultrafilter of subsets of X is principal every subset of X is a tile. Examples: every Hausdorff convergence space, every T1 quasi-sober topological space, every poset with the ascending chain condition under the Alexandroff topology.

Pointfree Convergence Relation to Stone duality

Outline

1 Introduction 2 The convergence space/focale duality 3 Sobrification 4 Relation to Stone duality 5 Conclusion

Pointfree Convergence Relation to Stone duality

And Stone duality?

Is there any connection between our L ⊣ pt adjunction and the Stone adjunction OStone ⊣ ptStone?

Pointfree Convergence Relation to Stone duality

And Stone duality?

Is there any connection between our L ⊣ pt adjunction and the Stone adjunction OStone ⊣ ptStone? In a convergence space X, a subset U of X is open iff for every x ∈ U, for every F → x, U ∈ F. Begs for the following (imagine L = PX): Definition (Open element) In a convergence lattice L, u ∈ L is open iff for every F ∈ FL, limL F ∧ u = ⊥ ⇒ u ∈ F.

Pointfree Convergence Relation to Stone duality

Open elements

Definition (Open element) In a convergence lattice L, u ∈ L is open iff for every F ∈ FL, limL F ∧ u = ⊥ ⇒ u ∈ F. Problem: opens are closed under finite infs. . . but not under sups (even just the finite ones) in general. Will be solved by restricting to tile-complete convergence lattices.

Pointfree Convergence Relation to Stone duality

Tile completeness

Definition A convergence lattice L is tile-complete iff every tile in pt L is of the form ℓ♯ for some unique ℓ ∈ L. Recall that ℓ♯ is always a tile. Proposition L X is tile-complete for every convergence space X. Hence L ⊣ pt restricts to an adjunction between Conv and the subcategory CCL ⊆ CL of tile-complete convergence lattices.

Pointfree Convergence Relation to Stone duality

Tile-completion

Definition For every L, L =def T pt L is a tile-complete convergence lattice, with lim

L F = limL pt L F = (limL F♭) ♯.

• L is in fact the free tile-complete convergence lattice.

Proposition (“tile-complete = spatial”) The following are equivalent: L is tile-complete; ǫL : L → L is bijective; ǫL : L → L is an isomorphism in CL.

Pointfree Convergence Relation to Stone duality

How this solves our problem with opens

Proposition Let L be tile-complete, and OL its subposet of open elements. OL is a frame finite infs and sups are computed in OL as in L ǫL restricts to an isomorphism between OL and OStone pt L. Shows that, in fact, OL is a spatial frame Proved by showing directly the last item The first two points are corollaries.

Pointfree Convergence Relation to Stone duality

Decomposing Stone, part 1

Recall that OL = {open elements of L}. O: CFoc → Loc is a functor, and fits into: Top

C

Conv

L

• ⊤

CFoc

pt

• O

Loc

where C maps every topological space to its underlying convergence space.

Pointfree Convergence Relation to Stone duality

Decomposing Stone, part 1

Proposition For every convergence space X, the open elements of L X = PX are the open subsets of X. Hence: Top

C

• OStone
• Conv

L

• ⊤

CFoc

pt

• O

Loc

commutes: O ◦ L ◦C = OStone.

Pointfree Convergence Relation to Stone duality

Free Boolean algebras on frames

For a lattice Ω, let BΩ be the free Boolean algebra over Ω. if Ω distributive, then Ω order-embeds into BΩ Proposition Given a frame Ω, BΩ is a convergence lattice, with limBΩ F = ¬ {u ∈ Ω | u ∈ F} then every element of Ω is an open element of BΩ Corollary Given a frame Ω, BΩ is a tile-complete convergence lattice.

Pointfree Convergence Relation to Stone duality

Decomposing Stone, part 2

Write B for the composite functor Frm

B

CL

CCL

Proposition There is an adjunction Frm : B ⊣ O : CCL. Hence (recall Loc = Frmop, CFoc = CCLop): Top

C

• OStone
• Conv

L

• ⊤

CFoc

pt

• ⊤

O

Loc

• B

Pointfree Convergence Relation to Stone duality

Decomposing Stone, part 3

Let M : Conv → Top be the topological modification functor. U open iff for every x ∈ U, if F → x then U ∈ F i.e., OStoneMX = O L X Proposition M ◦ pt ◦ B = ptStone Proof omitted, however note the available simplification: pt ◦ B = pt ◦T ◦ pt ◦B ∼ = pt ◦B: because X temperate iff X ∼ = pt L for some L iff X ∼ = pt TX Top

C

• OStone
• Conv

L

• ⊤

M

• CFoc

pt

• ⊤

O

Loc

• B
• ptStone

Pointfree Convergence Relation to Stone duality

The final nail?

Looks good? Top

C

• OStone
• Conv

L

• ⊤

M

• CFoc

pt

• ⊤

O

Loc

• B
• ptStone

Pointfree Convergence Relation to Stone duality

The final nail?

Looks good? Top

C

• OStone
• ⊥

Conv

L

• ⊤

M

• CFoc

pt

• ⊤

O

Loc

• B
• ptStone
• . . . No (not yet): leftmost adjunction is in the wrong direction.

Pointfree Convergence Relation to Stone duality

Would look better

Conv

L

• ⊤

M

• ⊣

CFoc

pt

• O
• ⊣

Top

C

• OStone
• ⊤

Loc

ptStone

• B
• Indeed OStone ◦ M = O ◦ L

But pt ◦ B ∼ = C ◦ ptStone is an open problem —equivalent to: pt BΩ is topological for every frame Ω Best I can prove for now: it is pretopological. And that is true, because left adjoints determine right adjoints uniquely (thanks to the audience for pointing this to me!)

Pointfree Convergence Conclusion

Outline

1 Introduction 2 The convergence space/focale duality 3 Sobrification 4 Relation to Stone duality 5 Conclusion

Pointfree Convergence Conclusion

Conclusion

Plenty of new notions: convergence lattices, tiles, etc. Tiles are a bit mysterious—but required. Temperate spaces X are exactly those of the form pt L Take L = convergence lattice of tiles TX Tee-totalers: X ∼ = pt L X iff every compact ultrafilter is principal iff every subset is a tile Tee-totaler ⇒ temperate ⇒ quasi-sober, with equivalence for topological spaces Almost perfect relation to Stone duality Top

C

• OStone
• ⊥

Conv

L

• ⊤

M

• CFoc

pt

• ⊤

O

Loc

• B
• ptStone

Pointfree Convergence Appendix

Lambda notation

Lambda notation would make that slightly clearer. Let ⊥ = {0 ≤ 1} ¬A = A → ⊥ is just PA FL = ¬∧L, monotonic finite-inf-preserving maps from L to ⊥ limL : ¬∧L → L monotonic pt L = ¬ptL, monotonic finite-inf-and-sup-preserving maps x such that x(limL x) = 1

♯ : L → ¬¬ptL,

ℓ♯ = λx.xℓ

♭ : ¬∧¬¬ptL → ¬∧L,

F♭ = λℓ.F(λx.xℓ)