Quotients of projective Fra¨ıss´e limits

Quotients of projective Fra¨ıss´ e limits

The pseudo-arc

For X a topological space, U , V open coverings of X , say U refines V if

∀U ∈ U ∃V ∈ V U ⊆ V .

A continuum is a compact connected metric space. It is non-degenerate if it has more than one point.

A continuum X is chainable if every open cover of X is refined by a

chain: an open cover {U 1 , . . . , U n } such that U i ∩ U j 6= ∅ ⇔ |i − j| ≤ 1.

This is equivalent to ask that ∀ε ∈ R ⁺ , ∃f : X → [0, 1] continuous and

onto such that ∀t ∈ [0, 1] diam(f ⁻¹ ({t})) < ε.

The pseudo-arc

For X a topological space, U , V open coverings of X , say U refines V if

∀U ∈ U ∃V ∈ V U ⊆ V .

A continuum is a compact connected metric space. It is non-degenerate if it has more than one point.

A continuum X is chainable if every open cover of X is refined by a

chain: an open cover {U 1 , . . . , U n } such that U i ∩ U j 6= ∅ ⇔ |i − j| ≤ 1.

This is equivalent to ask that ∀ε ∈ R ⁺ , ∃f : X → [0, 1] continuous and

onto such that ∀t ∈ [0, 1] diam(f ⁻¹ ({t})) < ε.

The pseudo-arc

For X a topological space, U , V open coverings of X , say U refines V if

∀U ∈ U ∃V ∈ V U ⊆ V .

A continuum is a compact connected metric space. It is non-degenerate if it has more than one point.

A continuum X is chainable if every open cover of X is refined by a chain: an open cover {U 1 , . . . , U n } such that U i ∩ U j 6= ∅ ⇔ |i − j| ≤ 1.

This is equivalent to ask that ∀ε ∈ R ⁺ , ∃f : X → [0, 1] continuous and

onto such that ∀t ∈ [0, 1] diam(f ⁻¹ ({t})) < ε.

The pseudo-arc

For X a topological space, U , V open coverings of X , say U refines V if

∀U ∈ U ∃V ∈ V U ⊆ V .

A continuum is a compact connected metric space. It is non-degenerate if it has more than one point.

A continuum X is chainable if every open cover of X is refined by a chain: an open cover {U 1 , . . . , U n } such that U i ∩ U j 6= ∅ ⇔ |i − j| ≤ 1.

This is equivalent to ask that ∀ε ∈ R ⁺ , ∃f : X → [0, 1] continuous and

onto such that ∀t ∈ [0, 1] diam(f ⁻¹ ({t})) < ε.

The pseudo-arc

A continuum X is decomposable if there are proper subcontinua Y , Z such that

X = Y ∪ Z . Otherwise it is indecomposable.

A continuum is hereditarily decomposable (hereditarily

indecomposable) if all of its non-degenerate subcontinua are.

The pseudo-arc

A continuum X is decomposable if there are proper subcontinua Y , Z such that

X = Y ∪ Z . Otherwise it is indecomposable.

A continuum is hereditarily decomposable (hereditarily

indecomposable) if all of its non-degenerate subcontinua are.

The pseudo-arc

The pseudo-arc is the unique, up to homeomorphism, hereditarily indecomposable chainable continuum.

The pseudo-arc is the generic continuum: in the Polish space of all

subcontinua of [0, 1] ^N , its homeomorphism class is dense G δ .

The pseudo-arc

The pseudo-arc is the unique, up to homeomorphism, hereditarily indecomposable chainable continuum.

The pseudo-arc is the generic continuum: in the Polish space of all

subcontinua of [0, 1] ^N , its homeomorphism class is dense G δ .

Fra¨ıss´ e limits

Let K be a class of L-structures. K has

HP hereditary property if: for all A ∈ K and for all finitely generated substructure B of A, B is (isomorphic to some element) in K . JEP joint embedding property if: ∀A, B ∈ K , ∃C ∈ K such that A, B

are embeddable in K .

AP amalgamation property if: ∀A, B, C ∈ K , ∀e : A → B, f : A → C embeddings, ∃D ∈ K , ∃g : B → D, h : C → D embeddings, such that ge = hf .

Theorem. (Fra¨ıss´ e) L a countable language, K a non-empty countable set of finitely generated L-structures having HP, JEP and AP. Then there is an L-structure D, unique up to isomorphism, such that:

I card(D) ≤ ℵ ₀

I up to isomorphism, K contains exactly the finitely generated substructures of D

I D is ultrahomogeneus: every isomorphism between finitely generated

substructures of D can be extended to an isomorphism of D

Fra¨ıss´ e limits

Let K be a class of L-structures. K has

HP hereditary property if: for all A ∈ K and for all finitely generated substructure B of A, B is (isomorphic to some element) in K . JEP joint embedding property if: ∀A, B ∈ K , ∃C ∈ K such that A, B

are embeddable in K .

AP amalgamation property if: ∀A, B, C ∈ K , ∀e : A → B, f : A → C embeddings, ∃D ∈ K , ∃g : B → D, h : C → D embeddings, such that ge = hf .

Theorem. (Fra¨ıss´ e) L a countable language, K a non-empty countable set of finitely generated L-structures having HP, JEP and AP. Then there is an L-structure D, unique up to isomorphism, such that:

I card(D) ≤ ℵ ₀

I up to isomorphism, K contains exactly the finitely generated substructures of D

I D is ultrahomogeneus: every isomorphism between finitely generated

substructures of D can be extended to an isomorphism of D

Topological structures

L a first order language.

A topological L-structures is an L-structure A such that:

I A is a zero-dimensional, compact, second countable space

I Interpretations of relation symbols are closed

I Interpretations of function symbols are continuous

An epimorphism of topological L-structures A, B is a continuous surjection ϕ : A → B such that

I R ^B = ϕ × . . . × ϕ(R ^A ) for any relation symbol R

I f ^B (ϕ(x 1 ), . . . , ϕ(x n )) = ϕf ^A (x 1 , . . . , x n ) for any n-ary function symbol f

An isomorphism is a bijective epimorphism

Topological structures

L a first order language.

A topological L-structures is an L-structure A such that:

I A is a zero-dimensional, compact, second countable space

I Interpretations of relation symbols are closed

I Interpretations of function symbols are continuous

An epimorphism of topological L-structures A, B is a continuous surjection ϕ : A → B such that

I R ^B = ϕ × . . . × ϕ(R ^A ) for any relation symbol R

I f ^B (ϕ(x 1 ), . . . , ϕ(x n )) = ϕf ^A (x 1 , . . . , x n ) for any n-ary function symbol f

An isomorphism is a bijective epimorphism

Projective Fra¨ıss´ e families

∆ a family of topological L-structures, is a projective Fra¨ıss´ e family if:

I ∀D, E ∈ ∆, ∃F ∈ ∆, ∃F → D, F → E epimorphisms

I ∀C , D, E ∈ ∆, ∀ϕ 1 : D → C , ϕ 2 : E → C epimorphisms,

∃F ∈ ∆, ∃ψ 1 : F → D, ψ 2 : F → E epimorphisms such that

ϕ 1 ψ 1 = ϕ 2 ψ 2

Projective Fra¨ıss´ e limits

∆ a family of topological L-structures, D a topological L-structure.

D is a projective Fra¨ıss´ e limit of ∆ if:

I (projective universality) ∀D ∈ ∆, ∃ϕ : D → D epimorphism

I For any clopen partition of D there is an epimorphism of D to a structure in ∆ refining it

I (projective ultrahomogeneity) ∀D ∈ D, ∀ϕ ¹ : D → D, ∀ϕ ² : D → D

epimorphisms, ∃ψ : D → D isomorphism such that ϕ ² = ϕ 1 ψ

Projective Fra¨ıss´ e limits

Theorem. A countable, non-empty class ∆ of finite topological L-structures has a projective Fra¨ıss´ e limit if and only if it is a projective Fra¨ıss´ e family.

Such a limit is unique up to isomorphism.

Construction of the limit

A fundamental sequence for ∆ is a sequence (D _n ) in ∆ with epimorphisms π n : D n+1 → D n such that:

a) ∀D ∈ ∆, ∃n, ∃ϕ : D n → D epimorphism

b) ∀n, ∀E , F ∈ ∆, ∀ϕ 1 : F → E , ϕ 2 : D n → E epimorphisms,

∃m ≥ n, ∃ψ : D m → F epimorphism such that ϕ 1 ψ = ϕ 2 π ^m _n , where π ^m _n = π m−1 · . . . · π n : D m → D n .

For ∆ a countable non-empty class of finite topological L-structures, the existence of a fundamental sequence is equivalent to being a projective Fra¨ıss´ e family.

The projective Fra¨ıss´ e limit D of ∆ is the inverse limit of the sequence D 0

π

← D 1 π

← D 2 π

← D 3 π

← . . .

Construction of the limit

A fundamental sequence for ∆ is a sequence (D _n ) in ∆ with epimorphisms π n : D n+1 → D n such that:

a) ∀D ∈ ∆, ∃n, ∃ϕ : D n → D epimorphism

b) ∀n, ∀E , F ∈ ∆, ∀ϕ 1 : F → E , ϕ 2 : D n → E epimorphisms,

∃m ≥ n, ∃ψ : D m → F epimorphism such that ϕ 1 ψ = ϕ 2 π ^m _n , where π ^m _n = π m−1 · . . . · π n : D m → D n .

For ∆ a countable non-empty class of finite topological L-structures, the existence of a fundamental sequence is equivalent to being a projective Fra¨ıss´ e family.

The projective Fra¨ıss´ e limit D of ∆ is the inverse limit of the sequence D 0

π

← D 1 π

← D 2 π

← D 3 π

← . . .

Linear graphs

Let L = {R}, R a binary relation symbol.

A linear graph is a reflexive, symmetric, connected relation such that each element has at most three neighbours (including itself) and, if there are more then two elements, exactly two of them have exactly two neighbours

• − • − • − • − . . . − • − • − •

Linear graphs

Let L = {R}, R a binary relation symbol.

A linear graph is a reflexive, symmetric, connected relation such that each element has at most three neighbours (including itself) and, if there are more then two elements, exactly two of them have exactly two neighbours

• − • − • − • − . . . − • − • − •

Theorem. (Irwin, Solecki) The class ∆ 0 of finite linear graphs is a projective Fra¨ıss´ e class.

Let (P, R ^P ) be the projective Fra¨ıss´ e limit of ∆ 0 . Then

I R ^P is an equivalence relation, whose classes have at most two elements.

I P/R ^P is a pseudo-arc.

This allowed Irwin and Solecki to recover some known and new properties

of the pseudo-arc.

Theorem. (Irwin, Solecki) The class ∆ 0 of finite linear graphs is a projective Fra¨ıss´ e class.

Let (P, R ^P ) be the projective Fra¨ıss´ e limit of ∆ 0 . Then

I R ^P is an equivalence relation, whose classes have at most two elements.

I P/R ^P is a pseudo-arc.

This allowed Irwin and Solecki to recover some known and new properties

of the pseudo-arc.

Question: What are the possible generalisations?

1.

I Change the class of finite topological structures

I Restrict the class of epimorphisms to some meaningful subclass

I Reformulate accordingly notions of projective universality and projective homogeneity

This has been succesfully undertaken by Irwin.

2. Study all possible quotients of projective Fra¨ıss´ e limits for projective

Fra¨ıss´ e families of finite topological structures in the language {R}, R a

binary relation symbol.

Question: What are the possible generalisations?

1.

I Change the class of finite topological structures

I Restrict the class of epimorphisms to some meaningful subclass

I Reformulate accordingly notions of projective universality and projective homogeneity

This has been succesfully undertaken by Irwin.

2. Study all possible quotients of projective Fra¨ıss´ e limits for projective

Fra¨ıss´ e families of finite topological structures in the language {R}, R a

binary relation symbol.

Question: What are the possible generalisations?

1.

I Change the class of finite topological structures

I Restrict the class of epimorphisms to some meaningful subclass

I Reformulate accordingly notions of projective universality and projective homogeneity

This has been succesfully undertaken by Irwin.

2. Study all possible quotients of projective Fra¨ıss´ e limits for projective

Fra¨ıss´ e families of finite topological structures in the language {R}, R a

binary relation symbol.

Note that if a projective Fra¨ıss´ e family contains a disconnected structure, then the projective Fra¨ıss´ e limit (D, R ^D ) contains non-empty clopen subsets invariant under R ^D . So if R ^D is an equivalence relation, D/R ^D is disconnected.

Thus it appears to exist a strong limitation in getting continua with approach 2:

Theorem.

1. If Γ is a projective Fra¨ıss´ e family and Γ ⁰ ⊆ Γ is such that

∀G ∈ Γ, ∃G ⁰ ∈ Γ ⁰ , ∃ϕ : G ⁰ → G epimorphism, then Γ ⁰ is a

projective Fra¨ıss´ e family, with the same projective Fra¨ıss´ e limit as Γ. 2. If ∆ is a projective Fra¨ıss´ e family of finite connected reflexive graphs

of unbounded diameter, then ∆ ⊆ ∆ 0 .

Note that if a projective Fra¨ıss´ e family contains a disconnected structure, then the projective Fra¨ıss´ e limit (D, R ^D ) contains non-empty clopen subsets invariant under R ^D . So if R ^D is an equivalence relation, D/R ^D is disconnected.

Thus it appears to exist a strong limitation in getting continua with approach 2:

Theorem.

1. If Γ is a projective Fra¨ıss´ e family and Γ ⁰ ⊆ Γ is such that

∀G ∈ Γ, ∃G ⁰ ∈ Γ ⁰ , ∃ϕ : G ⁰ → G epimorphism, then Γ ⁰ is a

projective Fra¨ıss´ e family, with the same projective Fra¨ıss´ e limit as Γ.

2. If ∆ is a projective Fra¨ıss´ e family of finite connected reflexive graphs

of unbounded diameter, then ∆ ⊆ ∆ 0 .

Theorem The compact metric spaces that can be obtained as quotients D/R ^D , where (D, R ^D ) is a projective Fra¨ıss´ e limit of a projective Fra¨ıss´ e family of finite topological {R}-structures with R ^D an equivalence relation are:

1. a Cantor space

2. a disjoint union of m singletons and n pseudo-arcs (m, n ∈ N, m + n > 0)

3. a disjoint union of n compact metric spaces X _i , where

I

X

= P

∪ S

j ∈N

Q

P

a pseudo-arc

I

Q

a Cantor space clopen in X

∪

Q

dense in X

In particular, the only continua that can be obtained this way are the

point and the pseudo-arc.

Theorem The compact metric spaces that can be obtained as quotients D/R ^D , where (D, R ^D ) is a projective Fra¨ıss´ e limit of a projective Fra¨ıss´ e family of finite topological {R}-structures with R ^D an equivalence relation are:

1. a Cantor space

2. a disjoint union of m singletons and n pseudo-arcs (m, n ∈ N, m + n > 0)

3. a disjoint union of n compact metric spaces X _i , where

I

X

= P

∪ S

j ∈N

Q

P

a pseudo-arc

I

Q

a Cantor space clopen in X

∪

Q

dense in X

In particular, the only continua that can be obtained this way are the

point and the pseudo-arc.

The spaces in the previous theorem can be obtained, for example, with the following projective Fra¨ıss´ e families:

1. (Cantor space): When there is no bound on the number of non-trivial components of the members in the family; for instance: Γ = class of all finite reflexive graphs

2. (m points and n pseudo-arcs): ∆ ^m _n = class of all reflexive graphs with n + m connected components of which m are trivial and n are linear graphs

3. (n pseudo-arcs with a sequence of clopen Cantor sets converging to

each one of them): ∆ ^∞ _n = class of all reflexive graphs having

arbitrarily many connected components, exactly n of which

non-trivial, each of which a linear graph

The spaces in the previous theorem can be obtained, for example, with the following projective Fra¨ıss´ e families:

1. (Cantor space): When there is no bound on the number of non-trivial components of the members in the family; for instance:

Γ = class of all finite reflexive graphs

2. (m points and n pseudo-arcs): ∆ ^m _n = class of all reflexive graphs with n + m connected components of which m are trivial and n are linear graphs

3. (n pseudo-arcs with a sequence of clopen Cantor sets converging to

each one of them): ∆ ^∞ _n = class of all reflexive graphs having

arbitrarily many connected components, exactly n of which

non-trivial, each of which a linear graph

The spaces in the previous theorem can be obtained, for example, with the following projective Fra¨ıss´ e families:

1. (Cantor space): When there is no bound on the number of non-trivial components of the members in the family; for instance:

Γ = class of all finite reflexive graphs

2. (m points and n pseudo-arcs): ∆ ^m _n = class of all reflexive graphs with n + m connected components of which m are trivial and n are linear graphs

3. (n pseudo-arcs with a sequence of clopen Cantor sets converging to

each one of them): ∆ ^∞ _n = class of all reflexive graphs having

arbitrarily many connected components, exactly n of which

non-trivial, each of which a linear graph

The spaces in the previous theorem can be obtained, for example, with the following projective Fra¨ıss´ e families:

1. (Cantor space): When there is no bound on the number of non-trivial components of the members in the family; for instance:

Γ = class of all finite reflexive graphs

2. (m points and n pseudo-arcs): ∆ ^m _n = class of all reflexive graphs with n + m connected components of which m are trivial and n are linear graphs

3. (n pseudo-arcs with a sequence of clopen Cantor sets converging to

each one of them): ∆ ^∞ _n = class of all reflexive graphs having

arbitrarily many connected components, exactly n of which

non-trivial, each of which a linear graph

Small Polish structures

A Polish structure is a pair (X , G ) where:

I G is a Polish group acting faithfully on a set X

I the stabilisers of all singletons are closed

This generalises the notion of a profinite structure.

Small Polish structures

A Polish structure is a pair (X , G ) where:

I G is a Polish group acting faithfully on a set X

I the stabilisers of all singletons are closed

This generalises the notion of a profinite structure.

For A ⊆ X , let G _A be the pointwise stabiliser of A.

Let ~ a ∈ X ^<ω and A, B ⊆ X finite. Let π A : G A → G A ~ a, g 7→ g ~ a.

~ a is nm-independent from B over A if π _A ⁻¹ (G A∪B ~ a) is non-meagre in G A

(= π ⁻¹ _A (G A ~ a)).

(X , G ) is nm-stable if there is no infinite sequence A 0 ⊆ A 1 ⊆ . . . of finite subsets of X and a ∈ X such that a is nm-dependent from A i +1

over A i .

Example. (Krupi´ nski) Let P be the pseudo-arc. Then (P, Homeo(P)) is

a small, not nm-stable, Polish structure.

For A ⊆ X , let G _A be the pointwise stabiliser of A.

Let ~ a ∈ X ^<ω and A, B ⊆ X finite. Let π A : G A → G A ~ a, g 7→ g ~ a.

~ a is nm-independent from B over A if π _A ⁻¹ (G A∪B ~ a) is non-meagre in G A

(= π ⁻¹ _A (G A ~ a)).

(X , G ) is nm-stable if there is no infinite sequence A 0 ⊆ A 1 ⊆ . . . of finite subsets of X and a ∈ X such that a is nm-dependent from A i +1

over A i .

Example. (Krupi´ nski) Let P be the pseudo-arc. Then (P, Homeo(P)) is

a small, not nm-stable, Polish structure.

For A ⊆ X , let G _A be the pointwise stabiliser of A.

Let ~ a ∈ X ^<ω and A, B ⊆ X finite. Let π A : G A → G A ~ a, g 7→ g ~ a.

~ a is nm-independent from B over A if π _A ⁻¹ (G A∪B ~ a) is non-meagre in G A

(= π ⁻¹ _A (G A ~ a)).

(X , G ) is nm-stable if there is no infinite sequence A 0 ⊆ A 1 ⊆ . . . of finite subsets of X and a ∈ X such that a is nm-dependent from A i +1

over A i .

Example. (Krupi´ nski) Let P be the pseudo-arc. Then (P, Homeo(P)) is

a small, not nm-stable, Polish structure.

A dendrite is a locally connected continuum that does not contain simple closed curves.

Let W be Wa˙zewski universal dendrite.

Then (W , Homeo(W )) is a small Polish structure. Problems.

I Is it nm-stable?

I Characterise dendrites D such that (D, Homeo(D)) is small.

A dendrite is a locally connected continuum that does not contain simple closed curves.

Let W be Wa˙zewski universal dendrite.

Then (W , Homeo(W )) is a small Polish structure.

Problems.

I Is it nm-stable?

I Characterise dendrites D such that (D, Homeo(D)) is small.

A dendrite is a locally connected continuum that does not contain simple closed curves.

Let W be Wa˙zewski universal dendrite.

Then (W , Homeo(W )) is a small Polish structure.

Problems.

I Is it nm-stable?

I Characterise dendrites D such that (D, Homeo(D)) is small.