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Pyknotic and condensed sets have been introduced recently as a convenient framework for working with topological rings/algebras/groups/modules/etc. Recently there has been much (justified) excitement about these ideas and the theories coming from them, such as Scholze's analytic geometry. (Small note: the difference between pyknotic and condensed is essentially set-theoretic, as explained by Peter Scholze here.)

On the other side, cohesion is a notion first introduced by Lawvere many years ago that aims to axiomatise what it means to be a category of "spaces". It has been developed further by Schreiber in the context of synthetic higher differential geometry (and also by Shulman in cohesive HoTT and by Rezk in global homotopy theory, to give a few other names in this direction).

Recently, David Corfield started a very interesting discussion on the relation between these two notions at the $n$-Category Café. The aim of this question is basically to ask what's in the title:

What is the precise relation between pyknoticity and cohesiveness?

Along with a few subquestions:

  1. (On algebraic cohesion) It seems to me that the current notion of cohesion only works for smooth, differential-geometric spaces: we don't really have a good notion of algebraic cohesion (i.e. cohesion for schemes/stacks/etc.) or $p$-adic variants (rigid/Berkovich/adic/etc. spaces). Is this indeed the case?
  2. (On the relevance of cohesion to AG and homotopy theory) Despite its very young age, it's already clear that condensed/pyknotic technology is very useful and is probably going to be fruitfully applied to problems in homotopy theory and algebraic geometry. Can the same be said of cohesion?
  3. (On "condensed cohesion") Cohesion is a relative notion: not only do we have cohesive topoi but also cohesive morphisms of topoi, which recover the former in the special case of cohesive morphisms to the punctual topos. Scholze has suggested in the comments of the linked $n$-CatCafé discussion that we should not only consider cohesion with respect to $\mathrm{Sets}$, but also to condensed sets. What benefits does this approach presents? Is this (or some variant of this idea) a convenient notion of "cohesion" for algebraic geometry?
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    $\begingroup$ The idea of using relative cohesion in this case is mentioned by Schreiber back here: nforum.ncatlab.org/discussion/5473/etale-site/…. $\endgroup$
    – David Corfield
    Commented Apr 5, 2020 at 10:54
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    $\begingroup$ @Sofia no, Urs has been working on matters related to M-theory these days (with great results!). And I don't think anyone has, else we would see some attention on it in eg the nCafé or the nForum (or it's published in some really obscure place). $\endgroup$
    – David Roberts ♦
    Commented Apr 7, 2020 at 6:44
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    $\begingroup$ Perhaps working out some consequences in the relative cohesion case would lead somewhere, such as ncatlab.org/nlab/show/differential+cohomology+diagram. $\endgroup$
    – David Corfield
    Commented Apr 7, 2020 at 10:17
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    $\begingroup$ For a case of relative cohesion over stacks on arithmetic schemes, see ncatlab.org/nlab/show/differential+algebraic+K-theory. $\endgroup$
    – David Corfield
    Commented Apr 7, 2020 at 19:32
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    $\begingroup$ @DavidRoberts Thanks for clarifying that! (By the way I agree with you: I've been following Fiorenza–Sati–Schreiber's program on M-theory and I'm finding it very interesting!) $\endgroup$
    – Emily
    Commented Apr 9, 2020 at 0:59

3 Answers 3

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The work on analytic geometry is all joint with Dustin Clausen!

Your main question seems a little vague to me, but let me try to get at it by answering the subquestions. See also the discussion at the nCatCafe. Also, as David Corfield comments, much of this had been observed long before: https://nforum.ncatlab.org/discussion/5473/etale-site/?Focus=43431#Comment_43431

  1. Yes, I think cohesion does not work in algebraic or $p$-adic contexts. The issue is that schemes or rigid-analytic spaces are just not locally contractible.

  2. Cohesion does not seem to have been applied in algebraic or $p$-adic contexts. However, I realized recently (before this nCatCafe discussion), in my project with Laurent Fargues on the geometrization of the local Langlands correspondence, that the existence of the left adjoint to pullback ("relative homology") is a really useful structure in the pro-etale setting. I'm still somewhat confused about many things, but to some extent it can be used as a replacement to the functor $f_!$ of compactly supported cohomology, and has the advantage that its definition is completely canonical and it exists and has good properties even without any assumptions on $f$ (like being of finite dimension), at least after passing to "solid $\ell$-adic sheaves". So it may be that the existence of this left adjoint, which I believe is a main part of cohesion, may play some important role.

  3. As I already hinted in 2, this relative notion of cohesiveness may be a convenient notion. In brief, there are no sites relevant in algebraic geometry that are cohesive over sets, but there are such sites that are (essentially) cohesive over condensed sets; for example, the big pro-etale site on all schemes over a separably closed field $k$. So in this way the approach relative to condensed sets has benefits.

All of these questions sidestep the question of why condensed sets are not cohesive over sets, when cohesion is meant to model "toposes of spaces" and condensed sets are meant to be "the topos of spaces". I think the issue here is simply that for Lawvere a "space" was always built from locally contractible pieces, while work in algebraic geometry has taught us that schemes are just not built in this way. But things are OK if instead of "locally contractible"(="locally contractible onto a point") one says "locally contractible onto a profinite set", and this leads to the idea of cohesion relative to the topos of condensed sets.

Let me use this opportunity to point out that this dichotomy between locally contractible things as in the familar geometry over $\mathbb R$ and profinite things as codified in condensed sets is one of the key things that Dustin and I had to overcome in our work on analytic geometry. To prove our results on liquid $\mathbb R$-vector spaces we have to resolve real vector spaces by locally profinite sets!

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    $\begingroup$ I just learned about your and Dustin Clausen's work on analytic geometry; it would be an understatement to say that it is amazing! Would it be ok to ask a question about it? From what I understand, this version of analytic geometry includes many other theories as special cases, such as smooth and complex manifolds as well as Berkovich spaces or schemes. Q: How is your and Clausen's theory related to other recent approaches to combining algebraic and p-adic geometries, such as e.g. Paugam or Poineau's work on global analytic geometry? $\endgroup$
    – Emily
    Commented Apr 9, 2020 at 1:01
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    $\begingroup$ Thanks for your excitement :-)! Let me also point to work of Ben-Bassat and Kremnizer. These works are generally based on Banach or (dual) Frechet algebras, and I would expect that all of these theories can be embedded into our category of analytic spaces. However, these theories are generally less expressive than adic spaces (and as is well-known, I really like those) as they cannot talk about the second component of a Huber pair $(A,A^+)$, while our notion of analytic spaces is able to handle adic spaces in a very clean way. $\endgroup$
    – Peter Scholze
    Commented Apr 9, 2020 at 15:42
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    $\begingroup$ Actually, when I gave my course on condensed mathematics I was quite amazed how abstractly developing the desired $6$-functor formalism for coherent duality was almost automatically leading one to rediscover (discrete) adic spaces, in particular the seemingly obscure conditions on $A^+\subset A$ -- an open and integrally closed subring of powerbounded elements. $\endgroup$
    – Peter Scholze
    Commented Apr 9, 2020 at 15:45
  • $\begingroup$ 'condensed sets are meant to be "the topos of spaces" '. What conclusion was reached as to pyknotic/condensed sets and being a Grothendieck/elementary topos? And then similarly for pyknotic/condensed anima/infinity-groupoids and Grothendieck/elementary infinity-toposes? $\endgroup$
    – David Corfield
    Commented Apr 10, 2020 at 13:27
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We have a case of relative cohesion used in an algebraic geometric setting discussed at the nLab. The entry for differential algebraic K-theory interprets

  • Ulrich Bunke, Georg Tamme, Regulators and cycle maps in higher-dimensional differential algebraic K-theory (arXiv:1209.6451)

via cohesion over the base $Sh_\infty\left(Sch_{\mathbb{Z}}\right)$, ∞-stacks over a site of arithmetic schemes.

See also Urs Schreiber's entry, differential cohesion and idelic structure, and arithmetic elements of his research proposal, Higher theta functions and higher CS-WZW holography.

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    $\begingroup$ This sounds very exciting; thank you for the references! $\endgroup$
    – Emily
    Commented Apr 9, 2020 at 0:59
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    $\begingroup$ Maybe it's worth pointing out that in this example the base topos is arithmetic, but that the relevant "locally contractible" objects are still smooth manifolds (if I understand this correctly), so it's of a somewhat orthogonal nature. $\endgroup$
    – Peter Scholze
    Commented Apr 9, 2020 at 15:34
  • $\begingroup$ Yes, I was interpreting sub-question 2 generously, "the relevance of cohesion to AG". $\endgroup$
    – David Corfield
    Commented Apr 9, 2020 at 15:58
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Recent work by Qi Zhu, Fractured Structure on Condensed Anima, is looking to approach condensed mathematics via Jacob Lurie's concept of a fractured structure, seen as "local cohesive structure", cf. Remark 2.20.

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