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Hereditarily indecomposable continua as generic mathematical structures

Adam Bartoš, Wiesław Kubiś

TL;DR

The paper develops a robust framework that unifies Fraïssé theory with inverse-limit constructions in the setting of MU-categories to study generic continua. It introduces approximate Fraïssé theory, the Brown back-and-forth tool, and a type-based classification via a type functor $T$ for circle-like continua, enabling direct, limit-based characterizations of generic objects such as the pseudo-arc and $P$-adic pseudo-solenoids. The authors rederive the pseudo-arc as a Fraïssé limit and realize every $P$-adic pseudo-solenoid as a Fraïssé limit of continuous circle surjections, while providing a complete classification of generic continua over full subcategories of connected polyhedra. They also show that the pseudo-arc is generic over dominating subcategories of Peano continua and that universal pseudo-solenoids are generic in appropriate circle-like settings, highlighting the broad applicability of MU-categories and approximate Fraïssé methods to topological dynamics and continuum theory. Collectively, the work delivers a unified, largely model-theoretic approach to generic structures in topology with concrete classifications and new Fraïssé-limit realizations for circle-like and arc-like continua.$

Abstract

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 purpose, we develop a new robust approximate Fraïssé theory in the context of MU-categories, a generalization of metric-enriched categories, suitable for working directly with continuous maps between metrizable compacta. Our framework extends both the classical and projective Fraïssé theories. We reprove the Fraïssé-theoretic characterization of the pseudo-arc and we realize every P-adic pseudo-solenoid as a Fraïssé limit of a suitable category of continuous surjections on the circle. Moreover, we show that, when playing the game with continuous surjections between non-degenerate Peano continua, the pseudo-arc is always generic, while the universal pseudo-solenoid is generic over all surjections between circle-like continua. This gives a complete classification of generic continua over full non-trivial subcategories of connected polyhedra with continuous surjections.

Hereditarily indecomposable continua as generic mathematical structures

TL;DR

The paper develops a robust framework that unifies Fraïssé theory with inverse-limit constructions in the setting of MU-categories to study generic continua. It introduces approximate Fraïssé theory, the Brown back-and-forth tool, and a type-based classification via a type functor for circle-like continua, enabling direct, limit-based characterizations of generic objects such as the pseudo-arc and -adic pseudo-solenoids. The authors rederive the pseudo-arc as a Fraïssé limit and realize every -adic pseudo-solenoid as a Fraïssé limit of continuous circle surjections, while providing a complete classification of generic continua over full subcategories of connected polyhedra. They also show that the pseudo-arc is generic over dominating subcategories of Peano continua and that universal pseudo-solenoids are generic in appropriate circle-like settings, highlighting the broad applicability of MU-categories and approximate Fraïssé methods to topological dynamics and continuum theory. Collectively, the work delivers a unified, largely model-theoretic approach to generic structures in topology with concrete classifications and new Fraïssé-limit realizations for circle-like and arc-like continua.$

Abstract

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 purpose, we develop a new robust approximate Fraïssé theory in the context of MU-categories, a generalization of metric-enriched categories, suitable for working directly with continuous maps between metrizable compacta. Our framework extends both the classical and projective Fraïssé theories. We reprove the Fraïssé-theoretic characterization of the pseudo-arc and we realize every P-adic pseudo-solenoid as a Fraïssé limit of a suitable category of continuous surjections on the circle. Moreover, we show that, when playing the game with continuous surjections between non-degenerate Peano continua, the pseudo-arc is always generic, while the universal pseudo-solenoid is generic over all surjections between circle-like continua. This gives a complete classification of generic continua over full non-trivial subcategories of connected polyhedra with continuous surjections.
Paper Structure (18 sections, 72 theorems, 34 equations, 3 figures, 1 table)

This paper contains 18 sections, 72 theorems, 34 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Summary of the structure and results
  3. Preliminaries
  4. Continua and crookedness
  5. Crookedness of inverse limits
  6. Construction of the pseudo-arc
  7. Crookedness factorization theorem
  8. Generic objects
  9. Approximate setting
  10. Fraïssé theory
  11. Key notions
  12. Obtaining a free completion and the pseudo-arc
  13. Fraïssé categories and ages
  14. Applications
  15. Circle-like continua and pseudo-solenoids
  16. ...and 3 more sections

Key Result

Lemma 2.4

A continuous map $f \colon X \to Y$ between Hausdorff compacta is crooked at a (not necessarily admissible) quadruple $\langle A, B, U, V\rangle$ if and only if it is crooked at $\langle A \setminus V, B \setminus U, U, V\rangle$. Hence, when showing that $f$ is crooked (or $ε$-crooked when $Y$ is

Figures (3)

  • Figure 1: Crookedness of a space $X = F_A ∪ H ∪ F_B$ at a quadruple $\langle A, B, U_A, U_B\rangle$.
  • Figure 2: The canonical crooked simplicial surjection $c_5$.
  • Figure 3: The zig-zag sequence in the back and forth construction.

Theorems & Definitions (203)

  • Definition 2.1
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.7
  • proof
  • Lemma 2.9
  • proof
  • ...and 193 more