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. This embedding can be realized by either Rips complexes or analogs of Roe's anti-Cech approximations of spaces. In this 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 epsilon > 0 there is m > k such that the bonding map from vertical bar K-k vertical bar to vertical bar K-m vertical bar has a contiguous approximation g:vertical bar K-k vertical bar -> vertical bar K-m vertical bar which sends simplices of vertical bar K-k vertical bar to sets of diameter at most epsilon. (C) 2011 Elsevier B.V. All rights reserved.