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The Category of Bratteli Diagrams
Published online by Cambridge University Press: 20 November 2018
Abstract
A category structure for Bratteli diagrams is proposed and a functor from the category of $\text{AF}$ 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 $\text{AF}$ algebras (and at the same time, of Glimm’s classification of $\text{UHF}$ algebras). It is shown that the three approaches to classification of $\text{AF}$ algebras, namely, through Bratteli diagrams, $\text{K}$-theory, and a certain natural abstract classifying category, are essentially the same from a categorical point of view.
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- Copyright © Canadian Mathematical Society 2015
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