THE CATEGORY OF BRATTELI DIAGRAMS

A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Brat- teli 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 alge- bras (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.