Projective Fra\"{i}ss\'{e} limits and generalized Wa\.{z}ewski dendrites

TL;DR

This paper explores the construction of generalized Ważewski dendrites through projective Fraïssé limits of finite trees, utilizing categorical methods to establish their topological and homogeneous properties.

Contribution

It introduces a categorical framework for constructing generalized Ważewski dendrites as Fraïssé limits, extending previous work on projective Fraïssé limits of trees.

Findings

Constructed many generalized Ważewski dendrites as projective Fraïssé limits.

Applied categorical approach to realize all such dendrites.

Proved homogeneity of countable dense sets of endpoints.

Abstract

We continue the study of projective Fra\"{i}ss\'{e} limits of trees initiated by Charatonik and Roe and we construct many generalized Wa\.{z}ewski dendrites as the topological realization of a projective Fra\"{i}ss\'{e} limit of families of finite trees with (weakly) coherent epimorphisms. Moreover we use the categorical approach to Fra\"{i}ss\'{e} limits developed by Kubi\'{s} to construct all generalized Wa\.{z}ewski dendrites as topological realizations of Fra\"{i}ss\'{e} limits of suitable categories of finite structures. As an application we recover a homogeneity result for countable dense sets of endpoints in generalized Wa\.{z}ewski dendrites.