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.
[Math] Looking for a book that discusses differential topology/geometry from a heavy algebra/ category theory point of view
abstract-algebrabook-recommendationcategory-theorygeometryreference-request
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The short answer is: it depends! To do differential geometry you don't really need category theory at all, and the same could (nearly) be said for some flavors of algebraic geometry. That said, some people (myself included) learn things best from a categorical standpoint. If you get excited whenever people mention universal properties, and are happiest defining things in terms of a functor that they represent, then starting with some category theory may be a good thing for you. In that case, I would recommend working through the first chapters of the classic Categories for the working mathematician. In particular, you want a solid understanding of limits, adjoint functors, and the relation between the two.
Now, if you don't even know group theory yet, starting with category theory is a bad idea. It would be best to start with some abstract algebra, using one of the standard texts.
It may be that you are a more normal mathematician for whom "categories first" or "algebra first" is not a good idea. In this case, if you are interested in differential geometry, the best thing to do would be to learn differential geometry, and only spend time on other topics as necessary.
Algebraic geometry can be almost entirely non-categorical, or hyper-categorical depending on what you are interested in doing. How much category theory you will need to know depends primarily on your own tastes in algebraic geometry.
I offer that differential geometry may be a much broader field than algebraic topology, and so it is impossible to have textbooks analogous to Switzer or Whitehead.** So, although it isn't precisely an answer to your question, these are the most widely cited differential geometry textbooks according to MathSciNet. I've roughly grouped them by subject area:
- Bridson and Haefliger "Metric spaces of non-positive curvature"
- Burago, Burago, and Ivanov "A course in metric geometry"
- Gromov "Metric structures for Riemannian and non-Riemannian structures"
- Kobayashi and Nomizu "Foundations of differential geometry"
- Lawson and Michelsohn "Spin geometry"
- Besse "Einstein manifolds"
- Abraham and Marsden "Foundations of mechanics"
- Arnold "Mathematical methods of classical mechanics"
- O'Neill "Semi-Riemannian geometry with applications to relativity"
- Wald "General relativity"
- Hawking and Ellis "The large scale structure of spacetime"
- Helgason "Differential geometry, Lie groups, and symmetric spaces"
- Olver "Applications of Lie groups to differential equations"
- Rabinowitz "Minimax methods in critical point theory with applications to differential equations"
- Willem "Minimax theorems"
- Mawhin and Willem "Critical point theory and Hamiltonian systems"
- Katok and Hasselblatt "Introduction to the modern theory of dynamical systems"
- Temam "Infinite-dimensional dynamical systems in mechanics and physics"
- Guckenheimer and Holmes "Nonlinear oscillations, dynamical systems, and bifurcations of vector fields"
- Hale "Asymptotic behavior of dissipative systems"
- Hirsch, Pugh, and Shub "Invariant manifolds"
- Giusti "Minimal surfaces and functions of bounded variation"
Of the metric geometry books (#1), BBI's book is good for self-study, while Gromov's book is nice to have around and open to random pages.
Kobayashi and Nomizu is a hard book, but it is extremely rewarding, and I don't know of any comparable modern book - I would disagree in the extreme with whoever told you to skip it. It is only aged in superficial ways, such as some notations. Lawson and Michelsohn's book is quite advanced, and K-N vol. 1 (at least) would be a prerequisite. It includes a chapter on the Atiyah-Singer index theorem.
Besse's book covers "special Riemannian metrics", including a review of Riemannian, Kahler, and pseudo-Riemannian geometry. It is more of a reference book, good to look through sometimes.
For classical mechanics, Abraham and Marsden is quite sophisticated, and it is necessary to have a solid geometrical footing (roughly K-N vol 1) before going into it; Arnold's book is more introductory and would probably be very nice for self-study.
The general relativity books in #5 are all introductory and pretty approachable.
I'm not so familiar with the books #6-9. Guckenheimer and Holmes seems very friendly.
Personally, I'd also recommend Chow, Lu, and Ni's "Hamilton's Ricci flow," the content of which is necessary to understand the proofs of the Poincare and geometrization conjectures. The first chapter is an excellent mini-textbook on "classical" Riemannian geometry, reaching just beyond introductory books like Do Carmo's.
** just to underline the point in the first sentence - there are only five general or algebraic topology textbooks (Hatcher, Spanier, Rolfsen, Engelking, and Kelley), four differential topology textbooks (Bredon, Hirsch, Milnor "Morse Theory", and Milnor-Stasheff) and two convex geometry textbooks (Schneider and Ziegler) as widely cited as the above differential geometry textbooks
Best Answer
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.