Relaxed Cech Cohomology, Emeralds over Topological Spaces and the Kontsevich Integral

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.