Algebraic Geometry Belongs to Sheaves

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 $\mathcal{C}$ 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.

Charles Siegel is currently a postdoc at Kavli IPMU in Japan. He works on the geometry of the moduli space of curves.
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16 Responses to Algebraic Geometry Belongs to Sheaves

1. Totally right. I mean, what happens when you consider the sheaf of rings of smooth functions on (open subsets of) a smooth manifold? Differential geometry!

2. Akhil Mathew says:

Or the “Cousin problems” in several complex variables.

3. Exactly. Sheaves are such a general tool that they pop up everywhere. And while it’s nice that AG really exploits them properly, it’s a shame that so few others do, even in subjects that are close by…even in cases where people know a bit about how useful sheaves are!

4. Barbara says:

Well, of course as an Italian it is my duty to say that Algebraic Geometry is just an extended attempt to understand projective varieties :-) but I actually like the idea of teaching both diff/coplx manifolds and schemes as special cases of locally ringed spaces (I just hope my colleagues will allow me to do so once).
Sheaves do show up in other contexts, and not only within mathematics. I spent some years in a very strong computer science department, and some of the theorists knew sheaves, topoi, and stacks way better than I do.

5. @Barbara

I’ve actually got a very Italian view of algebraic geometry myself, to be honest. Also, I would love to sit in on a class like that, just on locally ringed spaces and covering several types of running examples. If you do get to teach it, could you make lecture notes available somewhere?

And I’ve been hearing CS people talking more and more about categories, topoi and the like, but hadn’t heard that they use stacks (in the categorical sense, as opposed to the LIFO sense) and I’m kind of curious what they do with them.

6. Glad to see you back. I posted a link to this from my Facebook page, which has roughly 600 “friends” — many in Math or Sience.

7. On the topic of sheaves popping up wherever local and global properties meet, see the work by Rob Ghrist in the area:

http://www.math.uiuc.edu/~ghrist/talks/sheavesandsensors.pdf

8. I’m aware of Ghrist’s work, he’s actually down the hall from me, and I talk to his grad students fairly regularly.

9. Irene says:

One reason people are sceptical about sheaves is that too much emphasis is made on the formal presentation and not enough time is spent explaining where the usefulness of the concept lies. When the wheel was first invented, things were easier because people could see that the new invention is useful.
Perhaps the sheaf enthusiasts could come up with a piece of writing that explains the usefulness of sheaves. Not just say ‘sheaves occur everywhere’, that’s not useful. Ordinary homological algebra (no sheaves yet here, at least not explicitly) occurs everywhere too, but many people don’t use it because they don’t understand how to tell when homological methods might come useful and how to dress their problem in homological-algebraic clothing. Sounds just like the situation with sheaves to me… Just because a notion pops up a lot doesn’t mean that it’s a convenient notion to manipulate. It’s here that the effort of the experts is needed: to explain to the rest of the people the PRACTICALITIES of sheaves.

10. G. Rodrigues says:

Sheaves are immensely useful even when the local-to-global aspect is not the most important e.g. when the base space is totally disconnected. Such examples appear in analysis (example: ultraproducts), algebra (example: representations of general algebras as section spaces, especially rings), etc.

11. G S says:

It seems noncommutative differential geometers don’t care so much about sheaves on their noncommutative riemannian manifolds. Instead they seem to use something else .. perhaps Morita equivalence, etc?

12. I know, quite literally, nothing about noncommutative geometry of any form, so I really can’t say anything about what they use instead of sheaves. Presumably the problem there is the same as the problem with noncommutative rings in general, that localization doesn’t work out.

13. Sheaves have appeared in computer science: Goguen J. (1992). Sheaf Semantics for Concurrent Interacting Objects. Mathematical Structures in Computer Science 11(4):52 -57. I have been developing a Cech like cohomology for sheaves of schedules in systems called combinatorial systems which are overlapped non-commutative actions. A schedule is a polynomial in these systems with a time function (rather like a logarithm). These are developed as models for manufacturing systems and supply chains. There is a rich lode of mathematics here with e.g groupoid actions representing possible changes. Is this a mathematician’s fantasy? Maybe but the author is has had a career as a business analyst and software developer in manufacturing companies. Some of this is in a site http://integratedexpertise.org.

14. I love your colloquial discussion!

15. I believe that within a some decades that sheaves will be seen as major tool in the creation of functors (“comparisons”, “measures of structure”) on large systems. For example “Structures of systems 1. Cohomology of manufacturing and supply network-like systems” http://www.tandfonline.com/doi/full/10.1080/03081079.2014.888551.(Int.Jnl.Gen.Sys.)
The analysis of large information and engineering systems (as opposed to building them) has few good analytical tools. Sheaves and their various cohomologies are a way to “integrate” the local and global. This might require the development of (co)homology-like tools that are defined not on groups and rings but perhaps lattices or graphs – the “(co)homology” looking for special objects such as modular lattices or expander graphs. The concept of a Topos might be drafted for logics of such systems with Grothendieck topologies coming from special classes of subsystems. With society becoming more and more dependent on large-scale systems these sprawling logics are ripe for some serious analytical tools.