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

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