Right now I’m making some polishing-up edits to my thesis before posting it on the archive. In the meantime, here, as I suggested, are some comments on one of the talks I’ve seen in the last week or so at UWO. This is partly to help me remember them and have something to look back on besides my dubious notes. Naturally, all the following is my understanding of what went down, so anything wrong is presumably my fault – if you notice, tell me!
Michael Misamore’s talk last week was called “Galois Theory”, though the name “Grothendieck” came up much more than that of poor Evariste Galois. It was interesting to me because it looked at this classical subject from the point of view of schemes. I remember enjoying an algebraic geometry course I took when I was at McGill on the subject of schemes, but I haven’t thought about them much since then.
If you don’t know the idea of a scheme, it’s simple enough, though the details get tricky fairly fast – it’s a (locally ringed topological) space which looks, locally, like the spectrum of some commutative ring. This is sort of like how a manifold is a space which looks locally like (or , if it’s a complex manifold). The spectrum of the ring is a space whose points are the prime ideals of the ring. Case in point would be if the ring is , the polynomial ring in variables. Then the spectrum looks a lot like (with the Zariski topology and some extra “generic points” for each algebraic variety). Schemes also come with, for each open set, a ring of functions on it – all of these together make up the “structure sheaf” of the scheme.
(Philosophical aside: One reason I like the idea of schemes, despite not having thought about them in a while, is that the concept of generating a space as the spectrum of a ring is intrinsically satisfying if you believe that “space” should be a derivative concept anyway. There are various reasons (another coming attraction, maybe!) for thinking this should be the case in fundamental physics. So spectra of commutative rings are nice because they suggest that “spaces” are secondary concepts, just used for classifying information about a ring. Schemes generalize this the same way manifolds generalize Euclidean space. Noncommutative geometry generalizes in an orthogonal direction – taking noncommutative rings and applying intuitions from the study of spectra. Apparently there’s even a concept of noncommutative schemes. Now, I don’t know much about any attempts to use any of these ideas in physics – and I can more easily conceive of cases where these objects fill in for configuration spaces, rather than space, per se – but they do give some reassurance that at least space doesn’t have to be fundamental.)
Anyway, Michael’s talk was ostensibly about Galois theory. Classically, this has to do with field extensions, and the Galois groups of field extensions (i.e. groups automorphisms from the extended field to itself which fix the base field). The basic point, though, seems to be that a field is just a special kind of commutative ring, which has only the one ideal – namely the whole field, since you can divide by everything. So the spectrum of a field is a single point (in fact, this motivates the idea of a “geometric point”: if a point in a scheme is given by a map for a field , then a “geometric point” is given by a map for any ring ).
So: you can look at a field extension as a one single-point scheme sitting over another, in a way that has some group of automorphisms associated to it. That’s not so interesting (though I find it a bit hard to visualize what automorphisms of a point mean – presumably something to do with the structure sheaf). More interesting is to have a covering map – actually a <i>finite etale cover</i> – from one scheme to another (“base”) scheme, and the group of covering transformations (“Deck transformations”) of the covering scheme – that is, the ones that can’t be detected after you apply the covering map. This is a more general analog of the Galois group of a field extension.
You can then see pretty clearly an analog of one of the well-known issues in Galois theory – namely, the problem of finding the absolute Galois group of a field , which is the Galois group of the separable closure of , (some nice subfield of the algebraic closure) over … This corresponds to finding the group of covering transformations of the universal cover of a scheme. The problem is, there may not be a universal cover. An example of a lack of a universal cover (in a category where maps are by definition algebraic maps) would be the punctured complex line . Covers of this are given by maps , whose group of covering transformations is the cyclic group . A universal cover should have infinitely many sheets (since the fundamental group of the punctured line is the integers), but there is no covering map which does this. (The map is analytic, but not algebraic).
So with that setup, Michael went on to explain about pro-groups, pro-representable functors, and how they address this issue. For standard covering spaces over a space , there’s a representable functor which takes a covering space over and gives the fibre over a point. You can represent this as a functor since the points in the fibre over of the universal cover are reached by liftings of distinct paths from to itself.
In the case of schemes, you don’t have a universal cover, necessarily, but you do have a category of covers – in fact, a pro-object in the category of objects over , which is a nice sort of diagram. If there were a universal cover, it would be a limit of this diagram – a universal object with maps into every object in the diagram.
(“Pro-object” and “geometric point”, by the way, are both examples of a common stragegy: replacing a singular gizmo – an object or a point – by a suitable map into the place where the gizmo would live. A pro-object in is a map from some nice small category, giving the “shape” of the diagram, into ; an geometric point in is a map from , giving the “shape” of the “point”, into .)
The fact that there isn’t a limit for this pro-object in the category of schemes over is inconvenient, but the philosophy seems to be that one should just use the pro-object anyway. On top of this, there’s a concept of a “pro-representable” functor, which can be described in terms of a function from a pro-object.
So there’s an analog of the representablity of the functor which gives fibres from covers. It involves a functor , where is a category of “finite etale covers” of a scheme (apparently this is the right notion, though I don’t quite grok it), which when applied to a cover gives the set of (geometric) points in over the (geometric) point .
The theorem says that it’s representable by some pro-object in the category of covers. Namely, , which by definition is the limit . Since each of these is a set, you get a pro-object in Sets: and over HERE, the limit exists! The same sort of thing happens when you look at the Galois groups – i.e. the (finite, since the covers are finite) groups of covering transformations form a pro-object in – a pro-group. Again, you can take the limit, and get a profinite group.
One of the main lessons in all this seems to be that if something doesn’t exist, you approach it in the limit. When there’s no limit, you can just take the whole net of things which are approximating it, and deal with that directly. When you start mapping that net into various other realms (as when we map in to to look at fibres, or to look at Galois groups), sometimes the resulting diagram will have a limit, and you can then look at that, if you like. Somehow it reminds me of compactification…
Anyway, it’s just as well I went through all this stuff, because the next talk was: Joshua Nichols-Barrer – “Intro to Quasicategories”. These turned out to have a lot to do with stacks – and once again we’re into algebraic geometry and Grothendieck’s turf… Today, because I wanted to learn more about stacks for reasons of my own (coming attraction?) I had a somewhat lengthy meeting with Josh, which helped a lot, even if it didn’t much explain quasicategories…
More on that later.