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<p>We characterize the pseudo-arc as well as <i>P</i>-adic pseudo-solenoids (for a set of primes <i>P</i>) as generic structures, arising from a natural game in which two players alternate in building an inverse sequence of surjections. The second player <i>wins</i> if the limit of this sequence is homeomorphic to a concrete (fixed in advance) space, called <i>generic</i> whenever the second player has a winning strategy. For this purpose, we develop a new robust approximate Fraïssé theory in the context of <i>MU-categories</i>, 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 <i>P</i>-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.</p>

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Hereditarily indecomposable continua as generic mathematical structures

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  • 发表日期：

    2026年1月8日

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  • Adam Bartoš,
  • Wiesław Kubiś

摘要

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.