Asymptotic dimension of coarse spaces via maps to simplicial complexes 

It is well-known that a paracompact space X is of covering dimension at most n if and only if any map f:X->K from X to a simplicial complex K can be pushed into its n-skeleton K^(n). We use the same idea to characterize asymptotic dimension in the coarse category of arbitrary coarse spaces. Continuity of the map f is replaced by variation of f on elements of a uniformly bounded cover. The same way one can generalize Property A of G.Yu to arbitrary coarse spaces.