# How to upgrade Category Theory skills for Algebraic Geometry?

I am doing a second advanced graduate course in Algebraic Geometry, with Hartshorne as a textbook.

The skillset I am least satisfied with is the application of the Category Theory to Algebraic Geometry. The thing is that Category Theory itself, within the context of the course, seems quite trivial. The example solutions make not much use of it, and its use was marginal in the homework.

However, exam papers of previous years appear to require some relatively advanced and dense juggling of things like fibre products, exactness of functors, and limits -- and at least more advanced than I am prepared for. I seem to lack the required intuition to use these formalisms effectively. It seems that there is more emphasis on Category Theory in those exam problems than in Hartshorne.

I have tried doing problems in Category Theory as well as going over with pen and paper of more advanced problems relevant to the issue, but I don't appear to make much progress because although I may understand how a solution of a more advanced problem works technically, my intuition is not substantially improved.

I am actually a quite bright person (measured by practically everything else, anyway), and I think that there must be an optimal strategy to address the issue. Also, I think that the problem I am experiencing is encountered by many students studying Algebraic Geometry.

So I would like to ask for advice on a strategy to approach this problem. Perhaps there are specific references and/or problem sets that can be recommended.

• Hartshorne explains all the category theory you need in the text. – Matt Samuel Dec 12 '14 at 23:10
• @MattSamuel Yes, and I have actually looked up the relevant category theory in several places, including lecture notes from different courses, other textbooks on algebraic geometry, etc. The issue I have is with developing category theory intuition for problems in algebraic geometry, which make relatively advanced, dense, or otherwise less intuitive use of category theory. Algebraic geometry is a vast, complex, and relatively unintuitive field itself, and it is necessary to be quite flexible and intuitive in application of category theory to it. – Jake Dec 12 '14 at 23:28
• I really like the videos by the Catsters: youtube.com/user/TheCatsters – Seth Dec 12 '14 at 23:55
• @Matt: "Hartshorne explains all the category theory you need in the text." Really? Hartshorne hides category theory as much as he can (as if this was something obscure), even though category theory is really needed to understand some of the key notions and to understand properly the details of some arguments. – Martin Brandenburg Dec 12 '14 at 23:57

Nothing can be more fertile and instructive than a careful reading through the classic book Categories for the working mathematician by S. Mac Lane. I think the key concept is that of an adjunction and (closely related) of a representable functor, because it literally appears all over the place in algebraic geometry. One motivating example is that the direct image functor $f_*$ is right adjoint to the pullback functor $f^*$. Using that, you can deduce easily that $f^*$ commutes with colimits without having to delve into the specific construction of $f^*$ since it is a general fact that left adjoints preserve colimits. All these type of general facts are contained in Mac Lane's book. An alternative is the book Sheaves in geometry and logic by Mac Lane and Moerdijk, which focusses on algebraic geometry and logic. It has the advantage of explaining everything in great detail, especially the notion of a sheaf, but this may also distract you from algebraic geometry a little bit too much. For sheaf cohomology one needs quite a bit of homological algebra, for which Weibel's book is one standard reference, but I think that usually texts on algebraic geometry already explain this in full detail (except perhaps for spectral sequences, which I had only understood after reading Grothendieck's treatment in EGA). You might be also interested in functorial algebraic geometry, which offers a quite natural and easy definition of schemes, their morphisms, quasi-coherent modules etc., see here for some sources.