Hereditarily indecomposable continua as generic mathematical structures

TL;DR

This paper characterizes certain complex continua, like the pseudo-arc and P-adic pseudo-solenoids, as generic structures arising from a game-theoretic framework and develops a new Fra"issé theory in MU-categories for this purpose.

Contribution

It introduces a new robust approximate Fra"issé theory in MU-categories and applies it to classify generic continua, extending classical Fra"issé theories.

Findings

Pseudo-arc is always generic in the game setting.

Every P-adic pseudo-solenoid can be realized as a Fra"issé limit.

Complete classification of generic continua over subcategories of connected polyhedra.

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 Fra\"iss\'e 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 Fra\"iss\'e theories. We reprove the Fra\"iss\'e-theoretic characterization of the pseudo-arc and we realize every P-adic pseudo-solenoid as a Fra\"iss\'e 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.