Advice: Algebra and category theory for geometry?
I'm interested in learning a bit of geometry. To start I'm (slowly) working my way towards differential geometry via Lee's Introduction to Smooth Manifolds. But, later on, I'd also like to study some (real) algebraic geometry (I'm interest in how it interacts with optimisation).
I have a fair bit of exposure to analysis (real/complex/functional, topology, measure theory, probability theory, optimisation, PDEs, stochastic processes, etc.) but virtually none to algebra (only linear algebra) or category theory.
Last trimester I attended part of an introductory course on smooth manifolds (couldn't finish it because too many things were going on at the same time). In it the lecturer occasional discussed concepts from algebra (for example, groups) and category theory (for example, universal properties) and I felt that I was missing out.
To rectify this I've been reading Conceptual Mathematics: A first introduction to categories by Lawrence and Schanuel. However, I found this mo post which got me worried I might be going at this the wrong way round.
So my questions are:
With the end goal of acquiring a working knowledge of differential and algebraic geometry, how much algebra and/or category theory should one know?
What references would you recommend to achieve the above, is Lawrence and Schanuel's book a good start? Even if so, what else would you recommend?
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.