  Within the discrete gauge theory which is the basis of spin foam models, the
problem of macroscopically faithful coarse graining is studied. Macroscopic
data is identified; it contains the holonomy evaluation along a discrete set of
loops and the homotopy classes of certain maps. When two configurations share
this data they are related by a local deformation. The interpretation is that
such configurations differ by "microscopic details". In many cases the homotopy
type of the relevant maps is trivial for every connection; two important cases
in which the homotopy data is composed by a set of integer numbers are: (i) a
two dimensional base manifold and structure group U(1), (ii) a four dimensional
base manifold and structure group SU(2). These cases are relevant for spin foam
models of two dimensional gravity and four dimensional gravity respectively.
  This result suggests that if spin foam models for two-dimensional and
four-dimensional gravity are modified to include all the relevant macroscopic
degrees of freedom -the complete collection of macroscopic variables necessary
to ensure faithful coarse graining-, then they could provide appropriate
effective theories at a given scale.
