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

Pointfree Convergence Pointfree convergence Jean Goubault-Larrecq ∗ and Fr´ eric Mynard ∗∗ ed´ ∗ 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 : O Stone ⊣ pt Stone : Loc = Frm op : O Stone maps each topological space X to its lattice of open subsets pt Stone 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 F L be the set of filters on a (bounded) lattice L FP X 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 →⊆ FP X × 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: FP X → P X such that: (Triv.) x ∈ lim ˙ x (Mono.) lim is monotonic. Almost lends itself to a pointfree formulation: Replace P X 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 lim L : F L → 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 ∈ F L ′ , lim L ′ F ≤ ϕ (lim L ϕ − 1 ( F )). Defines a functor L : Conv → Foc = CL op : L ( X , → ) = ( P X , 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 ∈ F L such that lim L 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 ∈ (lim L F ♭ ) ♯ . For ℓ ∈ L , ℓ ♯ ∈ P pt L ˆ FP pt L = { x ∈ pt L | ℓ ∈ x } lim � P pt L “ x ∈ ℓ ♯ ⇔ ℓ ∈ x ” ♭ ♭ is Kowalsky sum : for F ∈ FP pt L , F L ♯ F ♭ ∈ F L = { ℓ ∈ L | ℓ ♯ ∈ F} � L lim 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 ∈ (lim L F ♭ ) ♯ . Defines a functor pt: Foc = CL op → Conv On morphisms ϕ : L ′ → L in CL , pt ϕ = ϕ − 1 . Theorem Conv : L ⊣ pt : Foc . Unit η X : X → pt L X : η X ( x ) = ˙ x Counit ǫ L : L → L pt L : ǫ L ( 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 : O Stone ⊣ pt Stone : Loc = Frm op induces a monad S = pt Stone O Stone 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 ∈ FP X 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 T 0 ”).

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 ♭ when X = pt L . Think of it as We shall impose further conditions on ϕ . The most important notion is that of a tile (wrt. ϕ )

Pointfree Convergence Sobrification Tiles ♭ if X = pt L ) Fix a map ϕ : pt L X → X ( ϕ = 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 T X of all tiles is a Boolean subalgebra of P X. In particular, in a sober topological space, every finite union of crescents is a tile. Take L = T X : can we make it a convergence lattice? Need to find lim L : 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 = (lim L 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 = (lim L F ♭ ) ♯ is a tile. Definition X is tiled iff convergence only depends on the tiles: if F ∩ T X = G ∩ T X , 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 = (lim L F ♭ ) ♯ is a tile. Definition X is tiled iff convergence only depends on the tiles: if F ∩ T X = G ∩ T X , 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 T 0 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 T X for L . Proof. Let L = T X . For F ∈ FP X , define lim L ( F ∩ L ) = lim F . Makes sense since X is tiled . Yields a tile since X is replete , so ( L , lim L ) convergence lattice. Define γ X : X → pt L by γ X ( x ) = ˙ x ∩ L . γ X surjective by Zorn’s Lemma + prime=maximal for filters on a Boolean algebra ( L = T X ) γ X injective since X is separated Finally, F → x iff γ X [ F ] → γ X ( 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: η X : X → pt L X is an isomorphism η X : 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 T 1 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 O Stone ⊣ pt Stone ?