  A notion of effective gauge fields which does not involve a background metric
is introduced. The role of scale is played by cellular decompositions of the
base manifold. Once a cellular decomposition is chosen, the corresponding space
of effective gauge fields is the space of flat connections with singularities
on its codimension two skeleton, ${\cal A}_{C-flat} \subset \bar{\cal A}_M$. If
cellular decomposition $C_2$ is finer than cellular decomposition $C_1$, there
is a coarse graining map $\pi_{C_2 \to C_1}: {\cal A}_{C_2-flat} \to {\cal
A}_{C_1-flat}$. We prove that the triple $({\cal A}_{C_2-flat}, \pi_{C_2 \to
C_1}, {\cal A}_{C_1-flat})$ is a principal fiber bundle with a preferred global
section given by the natural inclusion map $i_{C_1 \to C_2}: {\cal
A}_{C_1-flat} \to {\cal A}_{C_2-flat}$.
  Since the spaces ${\cal A}_{C-flat}$ are partially ordered (by inclusion) and
this order is directed in the direction of refinement, we can define a
continuum limit, $C \to M$. We prove that, in an appropriate sense, $\lim_{C
\to M} {\cal A}_{C-flat} = \bar{\cal A}_M$. We also define a construction of
measures in $\bar{\cal A}_M$ as the continuum limit (not a projective limit) of
effective measures.
