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.
Looking for a book that discusses differential topology/geometry from a heavy algebra/ category theory 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.
$\begingroup$ Do you suggest this book before the one with Görtz or your suggestion is to directly go to Görtz-Wedhorn? I have read commutative algebra (Atiyah-Macdonald) and Sheaf Theory (Tennison), both of which were suggested by you in different threads. I want to learn advanced algebraic geometry (from Görtz, hopefully by the time I read the first volume, the second one will show up) before moving on to my main aim, which is learning advanced-level p-adic arithmetic geometry, more specifically Mochizuki's p-adic Teichmüller theory. $\endgroup$– QGravityJan 22 at 14:21
1$\begingroup$ @QGravity I suggest going directly to the algebraic geometry book. As to the second volume I'm less optimistic than you: hell is paved with second volumes :-) $\endgroup$ Jan 22 at 16:34
I'd recommend Natural Operations in Differential Geometry by Kolár, Michor, and Slovák. Its approach is that of describing differential geometry via categorical constructions within the category of manifolds and via "bundle functors" on the category of manifolds. In particular, it has a beautiful categorical treatment of the Frölicher-Nijenhaus bracket of vector-valued differential forms, which a) is an awesome generalization of the Lie bracket of vector fields, and b) allows a very general treatment of the concept of connections on bundles.
To put the approach in context, it is I think complementary to the algebraic geometry-esque approach of Wedhorn's book that was also recommended. The latter is more interested in how manifolds sit in the category of locally ringed space and how categorical constructions on locally ringed spaces specialize to some classical constructions in the theory of manifolds, whereas Natural Operations in Differential Geometry starts with classical constructions of differential geometry and isolates their categorical formulations.
$\begingroup$ Very nice recommendation. $\endgroup$– user170039Jun 6, 2018 at 14:03
A few possible choices can be found in the noncommutative geometry literature, e.g. Gracia-Bondía, Varilly, and Figueroa's Elements of Noncommutative Geometry or Madore's An Introduction to Noncommutative Differential Geometry and its Physical Applications.
Noncommutative geometry is, in a sense, predicated on the fact that ordinary manifold theory can be characterized algebraically, and then generalized from that perspective, so there should be plenty for you to find there.
There is an extensive treatise with applications to many sciences (not just physics) titled ¨Applied Differential Geoometry, A modern Introduction¨ from Ivancevic.