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I am a fan of category theory in general, and I appreciate that various brands of generalized smooth spaces (Diffeological spaces, Frechet SpacesChen spaces, Frolicher spaces ...) form much nicer categories of spaces at the expense of having somewhat more convoluted objects. I might be interested in taking up the study of one form of generalized smooth space or another, but my conscience will not let me unless I see that they can actually buy me a more conceptual understanding of regular old manifolds.

So I would like a Big List of theorems about manifolds whose proof can be made significantly shorter or more conceptual by making use of generalized smooth spaces and maps between them. Something like a standard construction in the manifold setting becoming representable in the new setting, and this makes short work of some (previously) complicated theorem.
show/hide this revision's text 2 "conscience," not "conscious"

I am a fan of category theory in general, and I appreciate that various brands of generalized smooth spaces (Diffeological spaces, Frechet Spaces, ...) form much nicer categories of spaces at the expense of having somewhat more convoluted objects. I might be interested in taking up the study of one form of generalized smooth space or another, but my conscious conscience will not let me unless I see that they can actually buy me a more conceptual understanding of regular old manifolds.

So I would like a Big List of theorems about manifolds whose proof can be made significantly shorter or more conceptual by making use of generalized smooth spaces and maps between them. Something like a standard construction in the manifold setting becoming representable in the new setting, and this makes short work of some (previously) complicated theorem.
show/hide this revision's text 1 [made Community Wiki]

Nice application of generalized smooth spaces

I am a fan of category theory in general, and I appreciate that various brands of generalized smooth spaces (Diffeological spaces, Frechet Spaces, ...) form much nicer categories of spaces at the expense of having somewhat more convoluted objects. I might be interested in taking up the study of one form of generalized smooth space or another, but my conscious will not let me unless I see that they can actually buy me a more conceptual understanding of regular old manifolds.

So I would like a Big List of theorems about manifolds whose proof can be made significantly shorter or more conceptual by making use of generalized smooth spaces and maps between them. Something like a standard construction in the manifold setting becoming representable in the new setting, and this makes short work of some (previously) complicated theorem.