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There is another way to develop differential geometry without atlases, and even without charts, that is Diffeology. I'm not sure this is the right answer to your question but it worths looking at.

Comment: There are many ways to develop differential geometry without atlases. You may change the category of differentiable manifolds for a larger one. This is the case for Diffeology, or Differential Spaces à la Sikorski. These two approaches correspond to the two ways you may interpret the smooth structure: in the first case the smooth structure is characterized by the smooth parametrizations in the space (called plots), this is the "in-way", in the second case the smooth structure is characterized by the smooth functions from the space into the field of real numbers $\bf R$, the "out-way". In some sense they are "dual" but not equivalent approaches. The "intersection" of these two approches gives the so-called Frölicher spaces (reflexive diffeological spaces). The diffeological way gives a richer category than the Sikorski's one. For example it gives a non trivial structure for quotients (like spaces of leaves of a dense foliation, for example) where Sikorski structure is trivial. The two of them gives access to infinite dimensional spaces, without necessarily modeling these spaces on topological vector spaces. You can develop a whole theory of homotopy, cohomology, differential calculus and De-Rham cohomology, groups, fiber bundles etc. in diffeology without loosing much of what you have learned in manifold differential geometry.

Well, there is a lot say, may be too much for this discussion :-)

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Comment: There are many ways to develop differential geometry without atlases. You may change the category of differentiable manifolds for a larger one. This is the case for Diffeology, or Differential Spaces à la Sikorski. These two approaches correspond to the two ways you may interpret the smooth structure: in the first case the smooth structure is characterized by the smooth parametrizations in the space (called plots), this is the "in-way", in the second case the smooth structure is characterized by the smooth functions into the field of real numbers $\bf R$, the "out-way". In some sense they are "dual" but not equivalent approaches. The "intersection" of these two approches gives the so-called Frölicher spaces (reflexive diffeological spaces). The diffeological way gives a richer category than the Sikorski's one. For example it gives a non trivial structure for quotients (like spaces of leaves of a dense foliation, for example) where Sikorski structure is trivial. The two of them gives access to infinite dimensional spaces, without necessarily modeling these spaces on topological vector spaces. You can develop a whole theory of homotopy, cohomology, differential calculus and De-Rham cohomology, groups, fiber bundles etc. in diffeology without loosing much of what you have learned in manifold differential geometry.

Well, there is a lot say, may be too much for this discussion :-)

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There is another way to develop differential geometry without atlases, and even without charts, that is DiffeoogyDiffeology. I'm not sure this is the right answer to your question but it worths looking at.

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