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.

中文翻译：

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有序 BRATTELI 图的类别

摘要 提出了有序 Bratteli 图的范畴结构，其中同构与 Herman、Putnam 和 Skau 的等价概念一致。结果表明，康托极小系统范畴与简单有序布拉特利图范畴之间的自然一一对应实际上是范畴的等价。这给出了康托最小系统之间因子映射的 Bratteli-Vershik 模型。我们根据相应有序 Bratteli 图之间的合适映射（称为预同态）给出了康托极小系统之间的因子映射的构造，并且我们证明了通过这种方式获得了两个康托极小系统之间的每个因子映射。此外，解决一个自然的问题，