The local properties of structures are frequently studied by means of decompositions: the large structure is cut into (hopefully simpler)pieces whose properties are then studied together with the interconnections between pieces. Several decomposition schemes can be considered. For example, one can stress the regularity of the interconnections of the pieces as in modular decomposition of graphs (a recursive partition into modules such that, for each module, the neighborhoods outside the module of the vertices within the module are all equal [206, 266, 330]). As another example, one can consider a family of overlapping simple pieces covering the structure, in the spirit of the coherency between charts in topological atlases, such as the gluing axiom of sheaves  or the arrow construction for the category of labeled rooted forest and the category of labeled Feynman graphs .
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