TITLE: Want to learn differential geometry and want the sheaf perspective
QUESTION [14 upvotes]: 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.

REPLY [2 votes]: It is a bit late, but a book that fits perfectly with your demands was published after this questions was asked:

Name: "Manifolds, Sheaves, and Cohomology"
Author: Prof. Dr. Torsten Wedhorn, Department of Mathematics, Technische Universität Darmstadt, Germany.
Springer's Link: https://www.springer.com/gp/book/9783658106324

My opinion about this book is that the author makes a perfectly combination of abstract, topological and geometric flavor in a very gentle way that is not usually seen in advanced texts in differential geometry. Prerequisits are sumarized in the appendices and they are not very ambicious. Another plus point is that Prof. Dr. Torsten Wedhorn's research is focus in algebraic geometry (as far as I know), so you won't have any problems with the way he introduces the ideas.