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We generalize the theory of sheaves to chamber systems. We prove that, given a chamber system C and a family R of proper residues of C containing all residues of rank c1, every sheaf defined over R admits a completion which extends C. We also prove that, under suitable hypotheses, a sheaf defined over a truncation of C can be extended to a sheaf for C.I n the last section of this paper, we apply these results to a number of special cases.

Extending locally truncated chamber Systems by sheaves