There's a remark at 6:50 on the “little miracle” that the vertex and edge vector spaces, plus the map for each incidence, defines a sheaf. But I’m not sure I understand the miracle, and was hoping you might say a few more words about it. For instance, a graph appears simpler than real topological spaces (or even Zariski topologies :P) to me because it finite, at least locally. I'm assuming this is the cause of the miracle. Now for the next few videos, the sheaves we care about most are abstract varieties, which by definition come with a finite cover by affines. In the same way that we can build the whole sheaf on a graph from just the vertex/edge/incidence data, can we still leverage that kinda-finiteness to build the whole variety's sheaf from the affine parts and the restriction maps to their intersections? Or is the “kinda” not good enough, and we still have problems? Or am I missing the point with all this?
Not sure whether I can answer your question, but I give it a shot! Please complain if I didn’t quite get what you are asking for 😀 There is a important difference between a sheaf and a presheaf. The former has a locality condition build in, e.g. ‘continuous maps’ form a sheaf while ‘bounded maps’ do not. Essentially, sheaves are about creating bigger data from smaller data, while for presheafs this doesn’t work. In general, you should expect that a pre-sheaf is not a sheaf. The description of a sheaf on a graph really looks like what should be a presheaf on a graph. Now the miracle happens because there are only two types of open sets in a graph: ‘stars’ around vertices and ‘intervals’ for edges. The intervals do essentially nothing, so only the stars are left. But their open covers are also boring: they can only cover themselves. In other words, the topology on a graph is really simple (not just finite, but almost boring). Zariski is a weird topology, but there are still many nontrivial open sets, so we shouldn’t expect any miracle to happen. And indeed there is no free lunch 😂 For general varieties, patching sheaves together from their affine patches works, but its still more difficult than the graph case (again, because the graph case is topologically close to trivial). I hope that helps!
@@VisualMath Okay, that is very helpful. I was misunderstanding the miracle because our intuitions are different. For me, the surprise is that the description of a sheaf in the video is enough to define a sheaf on the entire graph; that is, on the various larger open sets. So I'm worried that we don't have enough data to define a presheaf, but your point is that we should be worried we already have too much independent data to define a sheaf! We're both right, of course. On one hand we've only made the most basic compatibility checks- the maps have the right domains and codomains- and usually that carelessness would kill our chances to be a sheaf. On the other, we actually *have not* given enough data to define a presheaf, since the lack of compatibility requirements makes for ambiguity on the larger open sets. Re: gluing: I guess the hope I had, made less vague, is that if we were just handed a finite poset (or at least some "admissible" one) describing how the affine patches (just the top-level sets) intersect, then we could play the graph game. That is, make independent choices for which varieties to label the poset elements with, define morphisms between the varieties with minimal checking aside from domain/codomain, and try to glue everything up. But now that I understand what miracle you're getting at, I see this is hopelessly optimistic, even combinatorially in 2+ dimensions. It's only possible for graphs because in their posets, elements are connected by at most one chain.
@@enstucky Excellent, that clarifies things for me as well. Indeed, our intuitions were just different - and that is totally fine 😀 And yes, I don’t think this has any chance beyond graphs.
I think of sheaves as presheaves following Yoneda’s lemma which also satisfy a covering condition. This covering condition describes which open sets are local and how to glue them together. This additional covering condition makes the presheaf into a sheaf. From the categorical perspective and following Yoneda’s lemma a presheaf describes the homomorphism into an object X ∈ C as Hom(-,X): C^op -> Set of an opposite small category C^op and category of sets Set. In this way the object X has a map X -> Hom(-,X) and can be understood by the morphisms (presheaves) into itself. See “Isabell duality” (2023) by Baez. The category of locally constant sheaves is equivalent to the category of covering spaces. See Example 1.2 in “Sheaves, covering spaces, monodromy and applications” (2016) by Calabrese.
I like to think of graphs as the easiest nontrivial structure where sheaves make sense. So sheaves on graphs are for me always the "baby example" I like to keep in mind. The categorical perspective then, as usual, works well if you already know what a sheaf is from examples in the wild 😀
@@VisualMath Good point. It is important to have examples and applications of a mathematical object. Your example of a cellular sheaf is interesting and natural to consider after working with matroids.
I have a silly question. For the graph + vector space example, is the (pre)sheaf just the assignment of vector spaces and the restriction maps between them? Or, is it somehow a valid assignment of elements in those vector spaces? I guess this last question can be phrased purely in terms of images of compositions of the restriction maps. I am asking since, from what I understand, sheaves are useful to show obstructions to certain things existing (which, from what I understand, is a very different motivation than in AG). As an example (which I will phrase informally as to maybe also help other people), let's say we want to show that there does not exist a line that I can draw on the mobius strip (where I don't distinguish the two sides) that is non-zero and locally constant. From what I understand, sheaf theory could be used to show that such a function does not exist. Is this done by showing that such a function is not a valid assignment according to the sheaf that one can construct on the mobius strip? If not, then what is done? Hopefully that makes sense. Thanks for your videos, they've always been very helpful and my go-to resource if I encounter something new.
The question is not silly at all! It assigns the whole vector space, not just elements, to vertices and edges. In general, a sheaf tries to associate “rich” data to open subsets: having a vector spaces is much better than just having a vector! Even better, with a vector space at hand, we can talk about maps, and they are the key players in all of this. Your Möbius strip example sounds like you have the following in mind (correct me if I am wrong): it is a line bundle over the circle that looks locally like a cylinder, but is not a cylinder globally. That is exactly the type of situation sheaves like. In AG a line bundle is often called “invertible sheaf”, and one can indeed use sheaf theory to prove the statement you mention. (Essentially an invertible sheaf of degree 0 has no non-zero sections unless it is the trivial sheaf.) So, yes, its exactly like you describe it. I would however say that using sheaf theory to prove the nonexistence of such a cut is a bit of an overkill 😅 Anyway, thank you for watching ☺
