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A category structure for Bratteli diagrams is proposed and a functor from the category of
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
algebras (and at the same time, of Glimm’s classification of
algebras). It is shown that the three approaches to classification of
algebras, namely, through Bratteli diagrams,
-theory, and a certain natural abstract classifying category, are essentially the same from a categorical point of view.
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