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nLab > Latest Changes: differential cohesion and idelic structure

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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 30th 2014

    There doesn’t seem to be a discussion for this page differential cohesion and idelic structure. Is this to be the general page for ’inter-geometry’?

    If so, it might be worth recording An Huang, On S-duality and Gauss reciprocity law, (Arxiv).

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 10th 2014
    • (edited Aug 10th 2014)

    Yeah, sorry, I had started the page expecting to finish it within a day, but then due to endless other distractions this didn’t happen. The page is not yet in a state that deserves to be announced. I am hoping to improve on this state of affairs the coming week.

    • CommentRowNumber3.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 11th 2014
    • (edited Aug 11th 2014)

    (posted this in the wrong thread)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 14th 2014
    • (edited Aug 14th 2014)

    Okay, I have now found time to work on bringing the entry differential cohesion and idelic structure a bit more into shape.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 15th 2014

    Here is a video recording of me speaking about this nLab entry at a workshop: video.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 18th 2014

    Interesting!

    In the natural correspondence between

    Π inf[Σ,BG]\Pi_{inf}[\Sigma, \mathbf{B}G] and [Π infΣ,BG][\Pi_{inf}\Sigma, \mathbf{B}G],

    you’re looking at just reductive groups isomorphic to their Langlands dual? Is that because above you’re considering GL nG L_n? How might dual groups appear naturally?

    Does this correspondence translate into arithmetic geometry as one would hope?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 18th 2014
    • (edited Aug 18th 2014)

    Ah, there should have been one L(−){}^L(-) in there. Have added it now.

    What that remark means to remark is that in terms of the de Rham space-functor Π inf\Pi_{inf} there is a curiuous similarity between

    1. “D-modules on the moduli stack of bundles”, which comes from Π inf[Σ,−]\Pi_{inf} [\Sigma, -] and

    2. “𝒪\mathcal{O}-modules on the moduli stack of local systems”, which comes from [Π infΣ,−][\Pi_{inf} \Sigma, -].

    It’s as if we are taking the Π inf\Pi_{inf} into the first argument of the internal hom. While, of course, dualizing the structure groups, too.

    I don’t know how a correspondence between the two would translates to arithmetic geometry, because I don’t have a particular insight into this correspondence, all I am observing here is that in terms of Π inf\Pi_{inf} the two legs of the would-be correspondence look curiously similar. I don’t know what this is hinting at, if anything. But the looks of it seems worth a remark.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2014
    • (edited Aug 19th 2014)

    By the way, if Σ\Sigma is reduced (as it is in the standard way it appears in Langlands theory) then Π inf[Σ,BG]≃[Σ,Π infBG]\Pi_{inf}[\Sigma,\mathbf{B}G] \simeq [\Sigma, \Pi_{inf} \mathbf{B}G].

    This is because for any UU

    H(U,Π ind[Σ,BG]) ≃H(ℜU,[Σ,BG]) ≃H(ℜU×Σ,BG) ≃H(ℜU×ℜΣ,BG) ≃H(ℜ(U×Σ),BG) ≃H(U×Σ,Π infBG) ≃H(U,[Σ,Π infBG]) \begin{aligned} \mathbf{H}(U, \Pi_{ind}[\Sigma,\mathbf{B}G]) & \simeq \mathbf{H}(\Re U, [\Sigma,\mathbf{B}G]) \\ & \simeq \mathbf{H}(\Re U \times \Sigma, \mathbf{B}G) \\ & \simeq \mathbf{H}(\Re U \times \Re \Sigma, \mathbf{B}G) \\ & \simeq \mathbf{H}(\Re (U \times \Sigma), \mathbf{B}G) \\ & \simeq \mathbf{H}(U \times \Sigma, \Pi_{inf}\mathbf{B}G) \\ & \simeq \mathbf{H}(U, [\Sigma, \Pi_{inf}\mathbf{B}G]) \end{aligned}

    So in view of this geometric Langlands asserts an equivalence between the 𝒪\mathcal{O}-modules on

    1. [Π infΣ,BG][\Pi_{inf} \Sigma, \mathbf{B}G]

    2. [Σ,Π infB LG][\Sigma, \Pi_{inf}\mathbf{B}{}^L G]

    On the one hand this looks weird, on the other hand it looks suggestive of something. If only I knew what that something would be.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 19th 2014

    That first XX is a Σ\Sigma, right?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2014

    Thanks for catching this. Fixed now.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2014

    We’re going to need a general abstract account of dual groups, no?

    What is this ’geometric Satake’ that seems important? I see nLab has a rather isolated page by Ben Webster – categorification via groupoid schemes – with a section on geometric Satake.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2014

    Small dabblings are never enough. It seems one should also know about ’Whittaker categories’. If only there were time, one could read through Gaitsgory’s outline and watch all the videos of the associated workshop. But it’s time for the day job.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2014

    We’re going to need a general abstract account of dual groups, no?

    Yes, we’ll need something. So far what we have is a general abstract account of all the ingredients for the statement except of that of Langlands dual groups. What is missing still is any general abstract insight of why the statement itself holds true. Of course that’s a tall order, too, and maybe there is no general abstract formulation (it would be somewhat shocking).

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2014
    • (edited Aug 19th 2014)

    Thanks for the pointer to the Satake isomorphism, that is useful. Will look into it, but need to run now.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2014
    • (edited Aug 19th 2014)

    I maded a mistake in the outlook at the end: the torsion approximation modality Π 𝔞\Pi_{\mathfrak{a}} is not quite monoidal, but “monoidal-except-possibly-for-respect-of-units”. Hence it descends from ∞\infty-modules not quite to E ∞E_\infty-rings, but to “E ∞E_\infty-rngs” (E ∞E_\infty-rings-without-unit). Have edited the text accordingly.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 19th 2014

    I thought ’rig’ meant without negatives, rather than without unit.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeAug 19th 2014
    • (edited Aug 19th 2014)

    Sorry, “rngs”. Thanks.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeAug 22nd 2014

    I have edited the text and section outline at differential cohesion and idelic structure a bit more, for readability and flow of the argument (or at least I hope that’s what I did).

    • CommentRowNumber19.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 23rd 2014

    inveranation?

    And presumably in the two commutative diagrams in example 3 the top entries should have Π dR rel\Pi^{rel}_{dR}.

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeAug 23rd 2014
    • (edited Aug 23rd 2014)

    Thanks!

    My fingers transmuted “incarnation” for some reason. (Actually I saw it yesterday, but then my battery died and I forgot about it. Thanks a whole lot for careful reading!)

    Regarding the rel{}^{rel}s: true, I have added them now where missing to example 3.

    But here there is a reason for omitting them, which of course the notes don’t discuss (yet), but let me say it:

    a) on reduced objects Π rel\Pi^{rel} and hence Π dR rel\Pi_{dR}^{rel} coincides with Π\Pi and Π dR\Pi_{dR} respectively. This is because we have a pushout

    ℜX ⟶ X ↓ ↓ Π(X) ⟶ Π relX \array{ \Re X &\longrightarrow& X \\ \downarrow && \downarrow \\ \Pi(X) &\longrightarrow& \Pi^{rel} X }

    and hence if the top morphism is an equivalence, meaning that XX is reduced, then Π relX≃ΠX\Pi^{rel} X \simeq \Pi X.

    b) This plays a role for the statement that EE-valued functions on Π dR relX\Pi^{rel}_{dR} X are “rational EE-valued functions”, namely functions X→♭ dREX \to \flat_{dR} E.

    Maybe I should find time to add discussion of this issue to the entry.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2014
    • (edited Dec 9th 2014)

    Just in case anyone watches the logs and is wondering:

    I have been edited a little the first section at differential cohesion and idelic structure, adding in a few more sentences here and there with a little bit more of explanation of what’s going on.

    (Because, re-reading this now, I found that section was more terse than necessary.)

    Also did some minor polishing edits in the remainder of the entry, some of them spilling over to fracture square.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeDec 9th 2014
    • (edited Dec 9th 2014)

    And re-reading section 2, i now found that the important definition of the relative cohesive modalities (the “infinitesimal” as opposed to “differential” modalities exhibiting cohesion over the base of formal moduli problems) was a bit hidden in the text. I have therefore now given them their own numbered definition just to make it stand out more and to be able to point to it more directly.

    This is in fact just the definition in the sheaf models considered in this section, not the general abstract one. The general abstract one is presently not on the nLab, but is briefly discussed in section 3.10.10 of dcct. This needs to be expanded on, eventually.

    (This is a point that we keep coming back to in other discussions and which we may want to come back to: while differential cohesion itself is not a level resolving cohesion, the infinitesimal cohesion that it induces IS. But I’ll better carry that discussion to the thread in Aufhebung.)

    • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 19th 2015

    In prop 11, where there is

    ♭ relΣ≃∐x∈ΣD x, \flat^{rel} \Sigma \simeq \underset{x \in \Sigma}{\coprod} D_{x},

    what would the corresponding Π dR rel(Σ) \Pi^{rel}_{dR}(\Sigma) be? It seems to want to be Σ\Sigma with all points removed.

    That’s like the imagery we heard on the Cafe once of spaces having each point removed:

    Spec C[z] is just the complex line C. As we start inverting elements of C[z], as we must do to make C(z), the effect on the spectrum is to remove bigger and bigger finite sets of points. The limit is where we remove all the points and we’re just left with some kind of mesh.

    So one can’t express the mesh directly, only through functions on it?

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2015
    • (edited Jun 25th 2015)

    Sorry for the slow reply here, somehow I almost missed this.

    Yes, certainly, that gives the space with all its points removed. There is exposition of this at the beginning of the entry here (discussed there for ease of presentation in the case that only one point is removed, but with ♭ rel\flat^{rel} we are just doing this for all points at once).

    So one can’t express the mesh directly, only through functions on it?

    That’s the picture in the presentation over an algebraic site, yes. But of course intrinsically in the differentially cohesive topos we are just forming some universal construction which is no more or less “directly expressed” than any other object here.

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