The Lelek fan and the Poulsen simplex as Fraïssé limits

Kubiś, Wiesław; Kwiatkowska, Aleksandra

doi:10.1007/s13398-016-0339-6

Cited by 7 publications

(6 citation statements)

References 17 publications

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“…without taking a quotient of a pre-space. In the same spirit, the second author and Kwiatkowska [30] realized the Lelek fan and the Poulsen simplex as Fraïssé limits. We should also mention that already Mioduszewski [37] constructed the pseudo-arc as a Fraïssé limit (without using that terminology) of special countable categories of copies of the unit interval and piecewise linear maps, and proved the surjective universality among arc-like continua.…”

Section: Introductionmentioning

confidence: 89%

“…Our theory of MU-categories and their Fraïssé limits can certainly be applied to more special topological categories, for example, categories of retractions between certain metric spaces (see e.g. [30]) or categories of homomorphic embeddings between certain topological algebras. In some situations one can work in categories enriched over metric spaces, which are special cases of MU-categories (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Hereditarily indecomposable continua as generic objects

Bartoš¹,

Kubiś²

2022

Preprint

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We characterize the pseudo-arc as well as P -adic pseudo-solenoids (for a set of primes P ) as generic structures, arising from a natural game in which two players alternate in building an inverse sequence of surjections. The second player wins if the limit of this sequence is homeomorphic to a concrete (fixed in advance) space, called generic whenever the second player has a winning strategy.For this aim, we develop a new approximate Fraïssé theory, in order to realize the above-mentioned objects (the pseudo-arc and the pseudo-solenoids) as Fraïssé limits. Our framework extends the discrete Fraïssé theory, both classical and projective, and is also suitable for working directly with continuous maps on metrizable compacta.We show, in particular, that, when playing with continuous surjections between non-degenerate Peano continua, the pseudo-arc is always generic. The universal pseudo-solenoid appears to be generic over all surjections between circle-like continua.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Hereditarily indecomposable continua as generic objects

Bartoš¹,

Kubiś²

2022

Preprint

Self Cite

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“…The class of standard finite-dimensional simplices n for n 2 N naturally forms a projective Fraïssé class in the sense of [30]; see [32]. The corresponding Fraïssé limit is the Poulsen simplex P .…”

Section: Choquet Simplices and Function Systemsmentioning

confidence: 99%

The Ramsey property for Banach spaces and Choquet simplices

Bartošová¹,

Lopez-Abad²,

Lupini³

et al. 2021

J. Eur. Math. Soc.

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We show that the Gurarij space G has extremely amenable automorphism group. This answers a question of Melleray and Tsankov. We also compute the universal minimal flow of the automorphism group of the Poulsen simplex P and we prove that it consists of the canonical action on P itself. This answers a question of Conley and Törnquist. We show that the pointwise stabilizer of any closed proper face of P is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simplex) is extremely amenable.These results are obtained via several Kechris-Pestov-Todorcevic correspondences, by establishing the approximate Ramsey property for several classes of finite-dimensional Banach spaces and function systems and their versions with distinguished contractions. This is the first direct application of the Kechris-Pestov-Todorcevic correspondence in the setting of metric structures. The fundamental combinatorial principle that underpins the proofs is the Dual Ramsey Theorem of Graham and Rothschild.

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“…The class of standard finite-dimensional simplices ∆ n for n ∈ N naturally form a projective Fraïssé class in the sense of [34]; see [38]. The corresponding Fraïssé limit is the Poulsen simplex P. Initially constructed by Poulsen in [67], P is a nontrivial metrizable Choquet simplex with dense extreme boundary.…”

Section: 1mentioning

confidence: 99%

The Ramsey property for Banach spaces and Choquet simplices, and applications

Bartošová

Lopez-Abad

Lupini

et al. 2017

Comptes Rendus. Mathématique

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Abstract. We show that the Gurarij space G and its noncommutative analog NG both have extremely amenable automorphism group. We also compute the universal minimal flows of the automorphism groups of the Poulsen simplex P and its noncommutative analogue NP. The former is P itself, and the latter is the state space of the operator system associated with NP. This answers a question of Conley and Törnquist. We also show that the pointwise stabilizer of any closed proper face of P is extremely amenable. Similarly, the pointwise stabilizer of any closed proper biface of the unit ball of the dual of the Gurarij space (the Lusky simplex) is extremely amenable.These results are obtained via the Kechris-Pestov-Todorcevic correspondence, by establishing the approximate Ramsey property for several classes of finite-dimensional operator spaces and operator systems (with distinguished linear functionals), including: Banach spaces, exact operator spaces, function systems with a distinguished state, and exact operator systems with a distinguished state. This is the first direct application of the Kechris-Pestov-Todorcevic correspondence in the setting of metric structures. The fundamental combinatorial principle that underpins the proofs is the Dual Ramsey Theorem of Graham and Rothschild.In the second part of the paper, we obtain factorization theorems for colorings of matrices and Grassmannians over R and C, which can be considered as continuous versions of the Dual Ramsey Theorem for Boolean matrices and of the Graham-Leeb-Rothschild Theorem for Grassmannians over a finite field.Recall that given a topological group G, a compact G-space or G-flow is a compact Hausdorff space X endowed with a continuous action of G. Such a G-flow X is called minimal when every orbit is dense. There is a natural notion of morphism between G-flows, given by a (necessarily surjective) G-equivariant continuous map (factor). A minimal G-flow is universal if it factors onto any other minimal G-flow. It is a classical fact that any topological group G admits a unique (up to isomorphism of G-flows) universal minimal flow, usually denoted by M (G) [18,33]. For any locally compact non compact Polish group G, the universal minimal G-flow is nonmetrizable. At the opposite end, non locally compact topological groups often have metrizable universal minimal flows, or even reduced to a single point. A topological group for which M (G) is a singleton is called extremely amenable. (Amenability of G is equivalent to the assertion that every compact G-space has an invariant Borel measure. Thus any extremely amenable group is, in particular, amenable.)The universal minimal flow has been explicitly computed for many topological groups, typically given as automorphism groups of naturally arising mathematical structures. Examples of extremely amenable Polish groups include the group of order automorphisms of Q [61], the group of unitary operators on the separable infinite-dimensional Hilbert space [30], the automorphism group of the hyperfinite II 1 factor and of in...

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The Lelek fan and the Poulsen simplex as Fraïssé limits

Cited by 7 publications

References 17 publications

Hereditarily indecomposable continua as generic objects

Hereditarily indecomposable continua as generic objects

The Ramsey property for Banach spaces and Choquet simplices

The Ramsey property for Banach spaces and Choquet simplices, and applications

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