This is the next in my series of posts on different ways to think about local systems. This time, we will consider an approach where we only build isomorphisms along infinitesimally short paths. If you pursued this line of thought long enough — and thought very hard about finite characteristic issues — you would come to the definition of a crystal. But I don’t plan to go nearly that far; I’ll just give you the intuitions in characteristic zero.
I think this is probably the hardest of the three perspectives I want to explore, but it logically comes second. Things will become a bit easier again when we get to our third perspective, connections.
To try to make this a bit easier, I’ll start with a nonstandard presentation; then reboot and give the actual definitions.
In our last discussion, we took a vector bundle on a space and, for every path in from to , we gave an isomorphism from to .
We don’t really need to know the maps for every path . We only need to know how to deal with short paths, because we can break any long path up into a chain of short paths.
Let’s try to make this notion of “short paths” precise. Let be an open subset of ; intuitively, will be the set of such that is near . We require that
(1) for every , the diagonal point is in .
(2) if and only if .
(3) for any , the set is a ball.
For example, if we have a metric on , we can take to be for some sufficiently small .
Definition B.1: An input of type B.1 consists of a (1) a vector bundle on and, for every , an isomorphism between and such that, if , and are all in , then .
It will turn out to be worthwhile to rephrase this definition in a way which avoids mentioning the individual points of . Let
We have three projections from to , call them , and . (So projects onto coordinates and and so on.) We have two projections from to : call them and .
So we can make
Definition B.2 An input of type B.2 is a vector bundle on , and an isomorphism of vector bundles on , such that .
That formula may look awfully abstract, but if you write it out you’ll see that it simply the formula from definition B.1.
Let’s try to see what looks like in local coordinates. For simplicity, I’ll take to be one dimensional. Let and denote the first and second coordinates on . So, for a section of , instead of writing and , I’ll write the more intuitive and .
Let be some (local) section of . We can expand as a power series
for some sections of , each depending on . Since has to be trivial on the diagonal, we have to have . There are many relations between the ‘s, which will come up in later posts.
By the way, if had dimension , we’d replace by . So there is nothing deep there.
Now comes the point that I think a lot of people will find challenging. Above, I worked with an honest open set in . What we are actually supposed to do is work with a formal neighborhood of the diagonal in . Intuitively, this means that we work with formal sums like (*). Let’s give a definition along these lines:
Definition B.3 An input of type B.3 is a vector bundle on , and, on an open cover of , formal power series like (*), which formally agree on overlaps, formally give linear maps and formally obey .
Inputs of type B.1 and B.2 are easily equivalent, but type B.3 is different. We don’t impose that our power series converge, and we identify functions that have the same Taylor series. (Such as and .) I believe that there is some theorem that we do get equivalent categories in the end, but this isn’t trivial.
In definition B.3, the spaces and went away. We can get them back, and prettify our definition, by using the notion of the formal neighborhood of the diagonal in . This requires the toolkit of modern algebraic geometry. Namely, let denote the sheaf of functions on . Let denote the subsheaf of functions vanishing on the diagonal. For every , there is a closed subscheme of , whose underlying topology is the same as , but where for any open set . The formal neighborhood of the diagonal, which I’ll denote by , is the
inverse direct limit of the . The space has two maps to , which we denote and . These are both the identity on the underlying topological space, but they pull back functions differently.
Definition B.4 An input of type B.4 is a vector bundle on , and an isomorphism of vector bundles on , such that .
This is nothing but a repackaging of definition B.3.
There is another mind blowing idea still to come, due to Grothendieck: is the universal infinitesimal thickening of . What this means is that, given an input of type B.4, a thickening of and two maps and from to , we canonically get an isomorphism between and . I think the -category people will really like this part! But this post is too long, so I’ll save it for another installment.
In the meantime, if you want to prepare for the connection perspective, you might want to think about the following:
Question: We want to be a linear map, so . What does this tell us about the ‘s? And, similarly, what does the condition say about the ‘s?