Is there a Book or a bunch of exercises to get used to diffeological spaces from a practical point of view? It seems to me that papers on this topic are mostly concerned with their very good categorical properties, but do little care about how well theses spaces behave in actuall calculations.
On ordinary manifolds I'm very used to local coordinates and doing actual calculations locally in the appropriate $\mathbb{R}^n$ or at least in terms of forms and vector-fields on it. As it appears to me, this locally nice behavior is the main advantage of a smooth manifold, in contrast for example to a bare topological manifold or even a topological space.
In addition I would say, this is the way differential geometry is mostly used in physics. Sure there are other situations like classification problems of things as topological insulators for example, but for a working physicist or even a well educated engineer, I think its mostly more the local data that counts.
So having an atlas is one of the better features of smooth manifolds and I would say that this is in fact the advantage against, lets say a more general sheaf theoretic approach. At least from an economic point of view.
What I mean is, that describing a manifold in terms of morphisms from a site into it, comes out as a huge amount of data, frequently as a proper class where in contrast the same manifold would only need finite many charts to describe it. So at least handling these data is very different in a sheaf theoretic approach vs. a 'chart-theoretic' approach.
Then on a first thought the 'chart-theoretic' approach appears smarter to me, beause it just let go of everything that is not important to the actual situation, whereas the sheaf-theoretic approach seems to be more 'brute-force-like' as it considers just everything.
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I know, this is a pretty vague question, but what I'm after is either some kind of atlas-theoretic approach to diffeological spaces or a bunch of exercise / a good explanation, how to deal with the sheaf theoretic approach and its huge amount of data in actual computations.
Likely its just me, but I think the sheaf theoretic approach is very Impractical.
