  The completely positive rank is an analogue of topological covering
dimension, defined for nuclear C*-algebras via completely positive
approximations. These may be thought of as simplicial approximations of the
algebra, which leads to the concept of piecewise homogeneous maps and a notion
of noncommutative simplicial complexes.
  We introduce a technical variation of the completely positive rank and show
that the two theories coincide in many important cases. Furthermore we analyze
some of their properties; in particular we show that both theories behave
nicely with respect to ideals and that they coincide with covering dimension of
the spectrum for certain continuous trace C*-algebras.
