  We introduce families of decorations of a same topological space, as well as
a family of sheaves over such decorated spaces. Making those families a
directed system leads to the concept of emerald over a space. For the
configuration space X_N of N points in the plane, connecting points of the
plane with chords is a decoration and the sheaf of log differentials over such
spaces forms an emerald. We introduce a relaxed form of Cech cohomology whereby
intersections are defined up to equivalence. Two disjoint open sets of X_N
whose respective points are connected by a chord is one instance of
intersection up to equivalence. One paradigm example of such a formalism is
provided by the Kontsevich integral.
