The category of ordered Bratteli diagrams

TL;DR

This paper establishes a categorical framework for ordered Bratteli diagrams, showing their equivalence with Cantor minimal systems and providing a model for factor maps and orbit equivalence in this setting.

Contribution

It introduces a category structure for ordered Bratteli diagrams, proving their equivalence with Cantor minimal systems and characterizing factor maps and orbit equivalence.

Findings

Category of ordered Bratteli diagrams is equivalent to Cantor minimal systems.

Constructs factor maps via premorphisms between diagrams.

Characterizes weak orbit equivalence through C*-algebra crossed products.

Abstract

A category structure for ordered Bratteli diagrams is proposed in which isomorphism coincides with the notion of equivalence of Herman, Putnam, and Skau. It is shown that the natural one-to-one correspondence between the category of Cantor minimal systems and the category of simple properly ordered Bratteli diagrams is in fact an equivalence of categories. This gives a Bratteli-Vershik model for factor maps between Cantor minimal systems. We give a construction of factor maps between Cantor minimal systems in terms of suitable maps (called premorphisms) between the corresponding ordered Bratteli diagrams, and we show that every factor map between two Cantor minimal systems is obtained in this way. Moreover, solving a natural question, we are able to characterize Glasner and Weiss's notion of weak orbit equivalence of Cantor minimal systems in terms of the corresponding C*-algebra crossed products.