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.
Best Answer
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.