  We consider the notion of dimension in four categories: the category of
(unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and
the category of (unbounded) separable metric spaces and (metrically proper)
uniform maps. A unified treatment is given to the large scale dimension and the
small scale dimension. We show that in all categories a space has dimension
zero if and only if it is equivalent to an ultrametric space. Also,
0-dimensional spaces are characterized by means of retractions to subspaces.
There is a universal zero-dimensional space in all categories. In the Lipschitz
Category spaces of dimension zero are characterized by means of extensions of
maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is
coarsely equivalent to a direct sum of cyclic groups. We construct uncountably
many examples of coarsely inequivalent ultrametric spaces.
