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I would like to learn some differential geometry: basically manifolds, differentiable manifolds, smooth manifolds, De Rham cohomology and everything else that is pretty much part of a course in differential geometry. I do however know some deal of category theory and algebraic geometry, and I would therefore like to learn differential geometry from a more "abstract" (categorical and algebraical) setting. Are there any good books for this? I was able to find a book called "Sheaves on Manifolds" but I don't know if it is a good book for learning the subject (AFAIK, the book might assume prior knowledge of differential geometry)

/edit/ Or just lecture notes.

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It sounds like you want to read this thread: mathoverflow.net/questions/14877/… –  Ryan Budney Mar 2 '11 at 22:03
    
Thanks Ryan. I think those (especially "abstract differential geometry") are a bit too extreme for me (but I have not had time to look through all comments, I will do that later). I still want the atlas experience, but I do not want the author to be afraid of the words "sheaf", "functor" and so on when such notation would ease things (for the readers who know category theory). So basically I am looking for some middle ground. The book Global Calculus might be what I want (I'll check it later), but I just wantet to put this out there. :) –  user7713 Mar 2 '11 at 22:15
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I don't think any authors are afraid of those words, there's just little reason to use them. Most authors try to minimize formal pre-requisites to a course. You're free to re-interpret any differential geometry text in whatever formalism you prefer. If there isn't sufficient benefit from a formalism, there's no reason to use it. –  Ryan Budney Mar 2 '11 at 22:38
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I would definitely not recommend Kashiwara-Shapira if you want to learn differential geometry. This is rather a book on microlocal analysis. If you want something general and abstract but with a good deal of differential geometry, I'd recommend emis.de/monographs/KSM/index.html where you can really learn some basic tricks of the trade and fundamental theorems, but it doesn't quite fit the bill in that sheaves are not mentioned that much, if at all. –  t.b. Mar 2 '11 at 23:00
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Out of curiosity, can someone provide some examples where categorical language simplifies the explanation of something in differential geometry? I can think of examples like the assignment of a manifold to its tangent bundle being a functor, but I do not find this very illuminating. –  Eric O. Korman Mar 3 '11 at 2:43

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I'm learning this stuff myself so take this with a large grain of salt but a commenter on this question suggested Warner, Foundations of Differentiable Manifolds and Lie Groups.

At a glance it looks like it goes through some of the usual topics but then does the de Rahm theorem using sheaves, so you might get along with it. Apart from Ch.5, though, I'm not sure how different it is from a standard treatment. It's a GTM book with minimal prereqs, and if you already know about sheaves it's probably a fairly gentle read.

I'd be interested to know how those in the know regard this text in relation to (what I take to be) the more usual textbooks.

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