Inverse sequences, rooted trees and their end spaces

TL;DR

This paper links inverse sequences to the coarse geometry of associated rooted trees, providing geometric characterizations of properties like Mittag-Leffler and new insights in shape theory.

Contribution

It introduces a geometric framework connecting inverse sequences with the coarse geometry of rooted trees, offering new characterizations and representations.

Findings

Category Tower-Set described via bounded coarse geometry

Mittag-Leffler property characterized by Lipschitz homotopy type

New shape morphism representations related to infinite branches

Abstract

In this paper we prove that if we consider the standard real metric on simplicial rooted trees then the category Tower-Set of inverse sequences can be described by means of the bounded coarse geometry of the naturally associated trees. Using this we give a geometrical characterization of Mittag-Leffler property in inverse sequences in terms of the Lipschitz, and metrically proper, homotopy type of the corresponding tree and its maximal geodesically complete subtree. We also some consequences in shape theory, in particular we obtain some new representations of shape morphisms related to infinite brunches in trees.