  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.
