Why is category theory the preferred language of advanced algebraic geometry?

Question

It's pretty easy to see how category theory could be applied in the case of abstract algebra, but doing so doesn't seem particularly useful, at the very least, for purposes of the stuff that one finds close to undergraduate abstract algebra.

So, how exactly does category theory become more relevant as a tool, and perhaps even a necessity, for formulating mathematical facts when we go into algebraic geometry?

Answer 1

I assume you understand how the basic language of category theory (morphisms, functors, natural transformation) is very convenient in many areas of mathematics, and that your question is more about why category theory seems to be particularly indispensable when it comes to modern algebraic geometry. For example, sheaves are often defined in introductory books on differential geometry without referring to category theory, and even Ext and Tor can be defined and studied with a bare minimum of categorical language. At what point in algebraic geometry does category theory become more than just a convenience?

I'd nominate the notion of a Grothendieck topology as one of the simplest concepts that is indispensable to modern algebraic geometry and that one cannot reasonably define without category theory. As others have noted, from an early stage, it seemed that the Weil conjectures were begging to be proved via cohomological techniques. But conventional topology was not up to the task of defining a suitable cohomology theory, and hence Grothendieck came up with the notion of a Grothendieck topology (and related concepts).

Since then, of course, increasingly sophisticated applications of category theory have entered algebraic geometry, but I think this is still a good example of a basic concept that requires more than a superficial use of categorical language.

Answer 2

$\quad$It is not difficult to show how category theory works on algebraic geometry by giving some examples, such as “the category of affine varieties is equivalent to the category of finitely generated k-algebras” somehow explains why algebraic geometry is based on commutative algebra; “maxSpec is not functorial, but Spec is” implies the motivation for the definition affine $n$-space and so on.
$\quad$Besides, I think there is another perfect way to make this problem understood——to study about Grothendieck.

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$\quad$Why Grothendieck’s theory succeeded? He himself attributed some of the reasons to the development of category theory. In fact, Serre’s discussion on coherent sheaf in FAC was very successful at first, but he mainly used the tools of etale space. This leads to some complicated and unclear proofs. However, Grothendieck in his EGA made clear that he won’t use etale space, but the language of category theory. This move make it possible to just talk about presheaf without etale space and sheaf, meanwhile contributing to some new discoveries in sheaf theory.
$\quad$And all this is for solving Weil Conjecture, Serre told Grothendieck that Weil Conjecture can be solved by establishing a cohomology theory on Weil’s “space”, in which Lefschetz Fixed Point holds. But the properties of Weil’s space is too bad to build a cohomology theory because it doesn’t has separation property.
$\quad$After a period of time, people came to understand the essence of sheaf cohomology and Cech cohomology by the tools from homological algebra. Then something like Grothendieck topology, even topos theory was invented by Grothendieck. And all this work are closely related to homology, whose foundation is totally category theory.

Answer 3

I feel that the question gives a slightly false impression. Since it has been reopened, let me explain my attitude, which I expect is not atypical among algebraic geometers. When I teach algebraic geometry, I usually spend part of a lecture discussing some basic category theory: categories, functors, limits, but not much more than that. (And most students are already familiar with this stuff from their algebra class.) So where does this come up?

1. In scheme theory (or in the study of varieties, for that matter), the (fibre) product is not the naive product of topological spaces, it's the categorical one. To prove existence it is necessary to know that the category of affine schemes and commutative rings are anti-equivalent. Similarly, group schemes are not groups in the usual sense, but are groups in the category of schemes.
2. When discussing operations on sheaves, it's good to know that $f^*$ is not the inverse of $f_*$ but the left adjoint. Also exactness properties of these operations are important, especially it comes time to discussing sheaf cohomology.
3. If I have time (but I usually don't), I might discuss what some moduli spaces are in terms of the functors they represent. So Yoneda's lemma is needed here.

Some parts of algebraic geometry (e.g. stacks, derived stuff) certainly use a lot more category theory and than what I described. But for much of it, a middling amount is enough.