what are some geometric moduli problems giving a map to the moduli of one-dimensional p-divisible groups? i know of modular curves and shimura varieties, but that's the extent of my knowledge
there are many variants but its roughly a vector bundle V (or more generally a G-bundle) on some F_p curve X with a meromorphic map F*V->V
and its natural because if you take the adelic description of shimura varieties as maximal compact\G(A)/G(K) and translate to function fields, shtukas are what you get (note that uniformization for Bun_G gives you the version over F_p bar, not F_p)
so is the analogy to dieudonn'e modules supposed to be something like the following
by dieudonn'e theory, p-divisible groups over an algebraically closed field k of characteristic p are classified by their dieudonn'e modules (which are exactly finite free W(k)-modules with a Frobenius and Verschiebung)
and i'm supposed to think of the dieudonn'e module of a p-divisible group as giving a vector bundle on the fargues-fontaine curve
well one has to be careful with the precise statement of the claim in that last message but it can be made precise
dhy weren't we just talking about this literally yesterday?
a drinfeld module is something like an affine bundle (a G_a-bundle) over your curve (whose function field you're interested in), together with some sort of compatible F_q[t]-action, and there's a natural way to get from drinfeld modules to shtukas by taking a fiberwise Hom(-, G_a); on the other hand, you can also take fiberwise p^{\infty}-torsion points to get your p-divisible group over your function field
@ArnavTripathy so it is true, then, that the mixed characteristic analogues of drinfel'd modules/shtukas/fancy-term-goes-here are dieudonn'e modules (or, i guess, dieudonn'e crystals)
Idiot question. What's the big deal about presenting, by categories with equivalences, \infty-categories? Is it just that people don't know whether the image of the simplicial localization spans all the \infty-categories or maybe something more subtle is going on?
@dhy What I was told a long time ago is that shtukas are analogous to local system by the function field-number field analogy and local systems in the p-adic case correspond to (over)convergent crystals (over a field they are just Dieudonné modules as noticed above) which are equivalent to p-divisible groups
@user40276 I'm not sure I understand your question. The reason people are interested in relative categories (i.e. categories equipped with a subcategory of weak equivalences) is that they show up in practice a lot
@user40276 that's helpful to know, thanks. this relates back to a question i had a while back: is the dieudonn'e module of a p-divisible group G actually the cohomology of some site associated to G?
to get the dieudonn'e module, you take the subgroup of primitive elements in lim H^1_cris(G[p^n]), but i don't know if this can arise as the cohomology of a site
@No. I mean why is it so difficult to present \infty-categories by categories with weak equivalences or model categories? Is it just because it's not known whether the essential image of the simplicial localization functor is the entire category of \infty-categories? Usually (for instance, in HoTT semantics) ...
@DenisNardin (Whoops! I forgot to write your name after the @) ...people restrict to locally presentable \infty-categories to get a 1-categorial presentation by Dugger's theorem, but I never tried to dig deep into the issue to see why it's so complicated to get a presentation of an arbitrary \infty-category
@user40276 There is a model structure on the category of relative categories (the Barwick-Kan model structure) that presents Cat_∞, and you can get relatively explicit presentations
But it's like giving a presentation by generators and relations of a group. Of course you can always do it: take all the elements of the group as a generator and the multiplication law as relations. This does not mean that working with a presentation is advantageous
In certain cases you have particularly useful presentations, and then you can use them but in general it's not advantageous using a 1-categorical presentation
For example you can take the category of simplices of an ∞-category and call an arrow an equivalence if it is sent to an equivalence by the last vertex map. This gives you a 1-categorical presentation of any ∞-category. In general it will be quite painful to work with though
@skd Hmm... overconvergent cohomology (rigid cohomology) and crystalline cohomology are a given by a Grothendieck topology. Maybe I didn't get your question...
@DenisNardin Thanksfor the answer. So I guess that the difficult is in getting a nice model category (satisfying things like properness) from a \infty-category.
For example, yes. Sometimes such a model category might not even exist. My favourite example is the Burnside category: such an important ∞-category, but it is way too small to be presented by a model category (it does not even have all finite colimits!)
@DenisNardin So you can always present by a category with weak equivalences, but not always by a model category? Is there any way to make this statement stronger? Like presenting by a category with fibrations or something like this...
