So this is just an attempt to make clear some things I’ve been thinking about lately in the areas of descent in algebraic geometry and topology. All of the following is well known and well documented in many places. I’ve particularly learned a great deal from Tyler Lawson about it, as well as the collective knowledge of the nLab, the Stacks Project and MathOverflow. I’ve also spent some time with Jacob Lurie’s various works, including DAG XI and section 6.2 of Higher Algebra. I’m probably also getting quite a bit incorrect, which is due to me and not any of the above mentioned people or references. Please send me an e-mail if there’s anything glaring, and I’ll attempt to correct it or make it rigorous. At the moment this is rather vague stuff…
Everyone’s pretty familiar with the (co)equalizer sheaf condition. That is, if we’ve got a map of rings and we want to know whether or not an -module has a (possibly unique) “representative” among -modules, we need to make sure that when we tensor up to being an -module along either the left unit or the right unit , we get the same module.
You might be more familiar with the opposite of this diagram that you get from thinking of such a module as a sheaf on . Obviously we’re leaving out some details here. If I’m working with, say, stacks (valued in groupoids) rather than sheaves, I’ll need to extend this diagram another level, which says that the cocycle condition isn’t satisfied up to equality, but rather to coherent isomorphisms (which then satisfy another condition up to equality!). If we’re working in a sheaf valued in, say, spaces or quasicategories, we have to extend this diagram all the way up (to ).
Now, I want to talk a bit about what we mean by a category of “descent data.” Generally for a map of “rings” (in whichever (discrete or quasi-)category we’re interested in) , the “descent problem” associated to this map is the question: Given an -module , when is it the case that there is an -module such that ? Again, in the case of modules, this is just the same thing as satisfying the sheaf condition. It’s saying that we’ve got some sheaf over and we want to “descend” it down to . Note that one typically leaves unsaid exactly how is an -module, since it’s assumed to be clear that it’s along the map . However, sometimes I’ll use the notation to indicate the precise way in which is an module, and perhaps more importantly, exactly in what way we’re lifting up the module structure on . Given the previous discussion, you can probably already see why this is going to be important to us.
The category of “descent data” for such a map should be, intuitively, things over with some kind of information on how to produce a thing over . There are many, many ways to formalize this. If you’re used to dealing with “covers” that look like , you’ll be thinking of descent data as something like “matching families” or gluing data. In that case, a descent datum is collection of objects over each element of the cover that agree on pairwise intersections. If by “agree” in the previous sentence we mean “are equal” then that’s all we need for descent datum. However, if by “agree” we mean “are isomorphic” (e.g. if we’re dealing with categories of modules or vector bundles or something rather than sets) then we have to throw in the “cocycle condition” which says that we can glue together these isomorphisms in the right way.
But if our “things over ” are spaces, then the isomorphisms between the restrictions won’t be equal, but rather isomorphic again. And we’ve got to glue those isomorphisms together in the right way. But there’s a rather nice way to frame this. Let’s take a map of rings , and start with an -module . We can glue along double intersections if, as we said above, the two possible ways of tensoring up to -modules are isomorphic, and these isomorphisms satisfy a cocycle condition. Now from here, there are three ways to tensor up to an -module. All of these ways need to be isomorphic, and these isomorphisms need to satisfy a cocycle condition one level up. And so this just keeps going, up and up and up. It turns out that there’s a really nice way to phrase all of this.
First we need the following:
Definition: Let be a cosimplicial ring, and be a cosimplicial module over . That is, is a functor and is a functor whose value in is . Then is said to be co-cartesian over if for every map in , the map induces an isomorphism
Let’s just unwind this definition for a second here. At level in the cosimplicial module , there are a whole bunch of maps for any other . What this condition is saying is the codomain of these maps is the same thing as just tensoring up to that cosimplicial level along the associated map in !
If you think back to what we talked about regarding sheaves of sets, you’ll see that at the first two levels of the cosimplicial diagram, this is the same as being a descent datum! Accordingly, we define the descent data for a map of homotopical rings to be the cosimplicial modules over the cosimplicial ring which are cocartesian, where we replace the isomorphisms in our definition of cocartesian with homotopy equivalences.
At the moment, what really interests me is the fact that this structure allows us to attempt to compute all possible descent data for a given -module along a map , and we do this by way of a Bousfield-Kan spectral sequence. Suppose we’re given an -module . Then what we’re interested now in computing are co-cartesian modules over the cosimplicial ring whose bottom level is the module . The requirement that this cosimplicial module be cocartesian is strong though, and the first bit of help it gives us is to tell us that , where there are copies of on the right hand side.
What’s really cool about this though is that it connects this notion of descent datum with the notion of a descent datum being a comodule over a canonical descent coring. If you’re not familiar with that, in the discrete case it says that for a map of rings , the category of descent data is equivalent to the category of comodules, where we regard as a coring, called the canonical descent coring. Sometimes it’s also called the Sweedler coring. It’s not hard to see the the data we’ve given above for being a co-cartesian module ends up being the same thing as being a comodule over in a homotopically coherent way. For instance, the first level of the cosimplicial module tells us that we have a map (well, two maps) , which is a coaction of the canonical coring on . The usefulness of the co-cartesian criterion is that we know that the two ways that can be isomorphic to are equivalent. In other words, descent data with a fixed base -module are the same thing as -comodules structures on . So how can we work out what possible such structures there are?
This is where the BKSS comes in. Notice that such a comodule structure on is going to be a system of maps from the constant cosimplicial object on , let’s denote it by , to the cosimplicial object which is just tensored (over !) with . So, to compute the space of descent data on , we need to attempt to compute the homotopy of the cosimplicial mapping space/spectrum/object of your favorite model category .
I hope to write more soon about how this space we’re computing can be compared to the space of twisted forms for along the map , and how if that map is a Galois or Hopf-Galois extension, the above computation actually has other interpretations in terms of the Galois group or the Hopf-Galois algebra of the extension.
P.S. - Please comment and let me know if anything above here seems off, or just plain wrong. I’d love to change it if so!