The category of Bratteli diagrams

TL;DR

This paper introduces a categorical framework for Bratteli diagrams, establishing a functorial connection to AF algebras and unifying various classification approaches.

Contribution

It develops a category structure for Bratteli diagrams and constructs a functor from AF algebras, unifying classification methods categorically.

Findings

Isomorphism of Bratteli diagrams matches Bratteli's equivalence.

A functorial formulation of AF algebra classification is achieved.

Different classification approaches are shown to be categorically equivalent.

Abstract

A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli's notion of equivalence, we obtain in particular a functorial formulation of Bratteli's classification of AF algebras (and at the same time, of Glimm's classification of UHF algebras). It is shown that the three approaches to classification of AF algebras, namely, through Bratteli diagrams, K-theory, and abstract classifying categories, are essentially the same from a categorical point of view.