Cubical sets have a directed homology, studied in a previous paper and consisting of preordered abelian groups, with a positive cone generated by the structural cubes. By this additional information, cubical sets can provide a sort of `noncommutative topology', agreeing with some results of noncommutative geometry but lacking the metric aspects of C* -algebras. Here, we make such similarity stricter by introducing normed cubical sets and their normed directed homology, formed of normed preordered abelian groups. The normed cubical sets NC_\theta associated with `irrational' rotations have thus the same classification up to isomorphism as the well-known irrational rotation C* -algebras A_\theta.
Keywords: Cubical sets, noncommutative C*-algebras, combinatorial homology, normed abelian groups
2000 MSC: 55U10, 81R60, 55Nxx
Theory and Applications of Categories,
Vol. 13, 2004,
No. 7, pp 114-128.