A Combinatorial Approach to Coarse Geometry

Using ideas from shape theory we embed the coarse category of metric spaces
into the category of direct sequences of simplicial complexes with bonding maps
being simplicial. Two direct sequences of simplicial complexes are equivalent 
if one of them can be transformed to the other by contiguous factorizations 
of bonding maps and by taking infinite subsequences. That embedding can be realized 
by either Rips complexes or analogs of Roe's anti-Cech approximations of spaces.

In that model coarse n-connectedness of K={K_1 -> K_2 -> ...} means 
that for each k there is m > k such that the bonding map from K_k to K_m 
induces trivial homomorphisms of all homotopy groups up to and including n.

The asymptotic dimension being at most $n$ means that for each k there is m > k
such that the bonding map from $K_k$ to $K_m$ factors (up to contiguity)
through an n-dimensional complex.

Property A of G.Yu is equivalent to the condition that for each k and for each e > 0 
there is m > k such that the bonding map from |K_k| -> |K_m| has a contiguous approximation 
g: |K_k| -> |K_m| which sends simplices of |K_k| to sets of diameter at most e.