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I should also point out that your

Your definition of a smooth manifold still uses atlases in a slightly disguised way because your definition it amounts to saying that every point of the topological a smooth manifold has is a neighborhood and topological manifold with an open cover whose elements are equipped with an isomorphism of the restriction of the structure sheaf to this neighborhood and the standard sheaf on R^n; the collection of all such neighborhood . This open cover is nothing else but an atlas.

Thus one still needs an atlas-free definition of a smooth manifold. One possible way to do this is to define the category of smooth manifold manifolds as the opposite category of the full subcategory of the category of real algebras consisting of real algebras satsifying some list of satisfying certain properties, e.g., the intersection of kernels of all homomorphisms to R is must be 0plus some additional conditions. One might hope that these additional conditions can be formulated in terms of dimensions of some vector bundles constructed from this algebra (e.g., tangent bundle, jet bundle, connections etc.).

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I should also point out that your definition of a smooth manifold still uses atlases in a slightly disguised way because your definition amounts to saying that every point of the topological manifold has a neighborhood and an isomorphism of the restriction of the sheaf to this neighborhood and the standard sheaf on R^n; the collection of all such neighborhood is nothing else but an atlas.

Thus one still needs an atlas-free definition of a smooth manifold. One possible way to do this is to define the category of smooth manifold as the opposite category of the full subcategory of the category of real algebras consisting of real algebras satsifying some list of properties, e.g., the intersection of kernels of all homomorphisms to R is 0 plus some additional conditions. One might hope that these additional conditions can be formulated in terms of dimensions of some vector bundles constructed from this algebra (e.g., tangent bundle, jet bundle, connections etc.).