Looking for a book that discusses differential topology/geometry from a heavy algebra/ category theory point of view So I am trying to learn differential geometry, stuff like manifolds, Lie groups, Stokes' theorem and so on. I have read about many books which discuss these topics, such as Lee's Smooth Manifolds and Tu's Introduction to Manifolds. But I have not seen any that is heavy in category theory or algebra. For example, Lee briefly introduces some basic category theory but it's only in a small section and he never mentions it again. I want a book that really uses category theory. The same with algebra. Most of these books try to minimize the algebra as much as possible. For example, Lee develops tensor products but I am pretty sure it is not in a very general/developed form. So is there a book which introduces differential geometry/topology from an algebraic / category theoretic point of view.
The book you're looking for is our friend @Wedhorn's Manifolds, Sheaves and Cohomology. 

Wedhorn's background is algebraic geometry, a subject in which he has already written (with his colleague Görtz ) a quite popular book, and his background shows in the book I recommend.
In Chapter 4 manifolds are presented as suitable ringed spaces i.e. topological spaces endowed with a  sheaf of rings (sheaves having been explained in chapter 3) and then the author introduces tangent spaces, Lie groups, bundles, torsors and cohomology.
This approach has been advocated since at least 50 years ago but Wedhorn's is one of the very rare books that consistently adopts the ringed space approach to manifolds.
The prerequisites in topology, categories, homological algebra and differential calculus are presented in appendices, so that the book is quite self-contained. 

Apart from its elegance and efficiency the ringed space approach to manifolds allows for a smoother (!) introduction to the more dificult theory of schemes (or analytic spaces) and is thus also an excellent investment for  ulterior study of more advanced material.
There is an extensive treatise with applications to many sciences (not just physics) titled ¨Applied Differential Geoometry, A modern Introduction¨ from Ivancevic.