We introduce anti-sheaves for Cr-manifolds and a category of anti-sheaves. The adjoint equivalence between the category of anti-sheaves and the category of manifolds is established. Using such an equivalence, we obtain a characterization for deciding whether two given manifolds arc Cr-diffeomorphic in terms of inherent W.(G)- sheaves. This provides the first known criterion for determining whether two given man- ifolds are diffeomorphic.
