Over at the n-Category Cafe, Tom Leinster has written an excellent post pointing out that sheaf theory is NOT a subfield of algebraic geometry. I feel I have a few things to add, not enough for a long post, but enough that I’d rather post here than fill up their comment thread (plus, the blatant cry for traffic, of course, as I am trying to bring this blog back to life).
So first off, I agree completely, sheaves are not owned by AG. In fact, I’d go so far as to say (in my completely reckless manner) that AG is a special case of the theory of sheaves, one where we can actually say a lot!
So first, what’s a sheaf? I’ve talked about this before, but let’s review quickly. Take a category, any category, and give it a topology, pick your favorite one. Then we define a sheaf of objects of a second category to be a contravariant functor satisfying a gluing condition in the topology on your category, spelled out in detail in my old post.
So, what’s a scheme? It’s just a sheaf in the Zariski topology on CommRing satisfying some conditions! (what they are is unimportant) So the general study of sheaves is far, far more general than the schemes. What about spaces? Well, same thing works! Stacks are a bit trickier, and I don’t understand them as well, but they’re also a purely categorical notion that is related to sheaf theory (some fuzziness due to 2-categories, I believe) and algebraic stacks are just particular ones satisfying some extra hypotheses.
So, I’ve said why algebraic geometry is owned by sheaves, but not what else is. As far as I can see, it’s pretty much the entirety of mathematics. Well…not all of it, but a ridiculously large swath. Sadly, most people in the other areas are terrified of sheaves or think “Oh, that’s just something those crazy AGfolk do” and never learn them seriously, even for the category of open subsets of a topological space.
Anyway, what’s a manifold? Well, it’s a topological space, equipped with a sheaf satisfying some conditions. Kind of like a scheme. This isn’t surprising, because they’re also an example of a locally ringed space, as are pretty much all other geometric objects of interest to most mathematicians. If you know much about the Mittag-Leffler problem, you know that sheaves are useful here, and the solution is to use the gluing axiom, so complex analysis has it’s hands in sheaves. Now, that’s not surprising either, considering its proximity to CAG.
A bit further from AG proper, but related to differential geometry a lot like how complex analysis is to CAG, we have DEs. Take a manifold, write down a differential equation on it, and what happens? People prove all sorts of local theorems, about existence and uniqueness and the like, then patch. It’s a sheaf! For any DE, there is a sheaf of solutions, and a global solution exists if you can patch one together to get a global section. This isn’t even the abstract categorical notion of a sheaf, it’s the “classical” one.
And that’s just the beginning. Sheaf theory is EVERYWHERE, and it’s a shame that so few people outside of AG really pick it up. So much of mathematics is of the form “if [local condition everywhere] then [global condition]” and the proofs are so often by patching and checking compatibility, sheaves are implicit, but forgotten.
Now, category theory began in algebraic topology, and has come to dominate, as a very useful language and bag of tricks, many areas of mathematics. It’s time that sheaf theory gets the same treatment properly, and becomes a standard part of the toolkit for everyone who does anything with local-global properties. And once that happens, I’m sure they’ll start popping up in surprising places.