The Rips complex at scale 
of a metric space 
is the abstract
simplicial complex whose simplices are the finite subsets of 
of diameter at most 
. This convention includes equality at the scale threshold.
Increasing 
gives inclusions of complexes, so a cycle present at one scale can become a boundary
at a larger scale. For example, three points at pairwise
distance at most 
span a filled triangle.
For a uniformly discrete metric space of bounded geometry, coarse K-homology is the direct limit of locally finite K-homology of these complexes. The coarse Novikov conjecture concerns an assembly map defined on that direct limit, rather than on only one complex at a fixed scale.
