I have removed the bulk of the entry, since that was a set of notes which has meanwhile evolved into polished form in arXiv:2008.01101, while what remained here had meanwhile become notationally inconsistent with both that and related $n$Lab entries, to the extent that it now would have been serious work to adjust notation.

So I just removed it for the time being, but added more prominent cross-links with the new entries *singular cohesion* and singular-smooth infinity-groupoid.

added brief commentary around the references by Schwede (suggesting regarding orbifold cohomology in global homotopy theory) and Juran (spellign it out)

]]>Peter Scholze has replied to my MO question. That’s right my response that a stable object in a $\infty$-topos that is relative cohesive over condensed $\infty$-groupoids (or pyknotic ones anyway) should generate a fracture square?

]]>Ok, with your encouragement I’ll give it a go.

]]>Yes, that would be the idea of “condensed cohesion”, that much of algebraic/arithmetic geometry could be given a cohesive home.

In good part due to your prodding, there is now a good candidate approach for how to attack this, which is what your MO question alludes to where it says

it appears that the $\infty$-topos of $\infty$-sheaves over the pro-étale site on all schemes over a separably closed field k is cohesive over $Sh_\infty(ProFinSet)$

But the number of people who are aware that it thus appears may be no larger than 3, and of those not a single one seems to be actively pursuing this.

If the hope is to find somebody else who might pick up this thread, then a first step to take might be to write out the expected statement cleanly, collect the available evidence for it and adorn it with some enlightning commentary on what it should be good for.

We should have an $n$Lab entry on this, maybe with the title_condensed cohesion_. It need not be long in the beginning, just the basic pointers that you have been mentioning in various comments.

]]>Well, it may have planted a seed in someone’s mind. An odd thought though that the synthetic approach to geometry that you have devised is just sitting there waiting for people to plug in different values. Could it be that people are missing out on really quite straightforward paths to important constructions?

You read something like

In this section, we construct a canonical $q$-deformation of de Rham cohomology: given a formally smooth $\mathbf{Z}_p$-scheme $X$, we construct a ringed site – the $q$-crystalline site of $X$ – whose cohomology yields a deformation of the de Rham cohomology of… (Prisms and Prismatic Cohomology)

Shouldn’t we expect to be able to provide it a modal cohesive home?

]]>Maybe it might help readers if this MO question could be focused a bit more:

The statement towards the end of the question – after “it appears that” – hints at what is, supposedly, the crucial point where cohesive meets condensed. But I worry that the set of readers who do all of: read to this point, pick up this info, unwind what it’s really saying, realize that this is the starting point for the whole question, and then have something to say about it might be tiny.

If you have time and energy to go after this, I’d suggest to first of all write a paragraph or two with a clean exposition of the idea behind “it appears that”. This would naturally go into an nLab entry “condensed cohesion”.

Then one could point people there to establish common ground from which on to start musings as in your MO question!

]]>I thought I’d try out an MO question on prospects for working out the consequences of cohesion over $Sh_{\infty}(ProFinSet)$. No replies yet.

]]>Perhaps the clearest expression of how he sees cohesion fitting in with condensed mathematics is in this MO answer.

So just replace $\infty-Grpd$ as base by $Sh_{\infty}(ProFinSet)$, and looks at all varieties of cohesion relative to that.

]]>Yes, that’s right!

In “Proper Orbifold Cohomology” we pick the base of the singular cohesion to be smooth cohesion (i.e. modeled on smooth manifolds) because that makes contact to the most widely understood default sense of orbifolds as smooth manifolds with singularities.

But already the second most widely understood default sense of *orbifold*, namely as Deligne-Mumford stacks in algebraic geometry, will – if it fits into cohesion at all – probably need some kind of “condensed”-style cohesive base $\infty$-topos, as we have been discussing numerous times in the past (in fact last time we discussed this you reported that it seemed as if Peter Scholze said he had checked what needs to be checked here, but i don’t know).

I chipped in at MO with a comment on using singular cohesion in the condensed setting. That is right, isn’t it, that one can just replace the smooth base $\infty$-topos by another? And, in general, all forms of cohesion (singular, infinitesimal, differential, elastic, etc.) may be understood in a relative sense, relative to any base?

]]>Thanks! Have fixed it (here).

]]>A few more typos:

]]>’cohesive” (unbalanced quote marks); supercede (’s’ rather than ’c’); DTopologcalSpaces; disivion; presverves

Okay, I’ll try to produce a skeleton of a project file, and then we can maybe see how to proceed.

]]>I see. What happens if you edit the file while on a plane/train/etc, or simply do not have Internet? Sometimes I would edit my files without Internet access.

In principle, my git setup is extremely easy to use: one runs

git sync

before and after editing the file, and git takes care of the rest. This is roughly analogous to your checking for any activity in the last minutes, I guess, but is more robust and convenient. In particular, it performs merges automatically, without any manual intervention (the only exception is when two or more people edit the same line in the file).

But using Dropbox could also work, if this is difficult or inconvenient for you.

]]>With Dropbox we just check if we see any activity in the last minutes, and we stay in touch by email.

I find it quick and easy, but if you explain what else to do, we can try something else.

]]>Re #87: I think pretty much any collection of examples will be original, since there are almost no examples in the literature. I think Simpson and Teleman compute essentially a single example of K(A,n) for an abelian Lie group A in their paper on the de Rham stack functor, i.e., the elastic setup. And my draft computes a bunch of examples for the shape functor, but that’s just a tiny part of the whole picture.

Concerning Dropbox, how do you solve the problem of multiple people editing the same file? (My coauthors use git for collaborative editing, which has built-in tools for automatic merging.)

]]>I’ll need to think about it more, but at the beginning of the article I imagine material as in chapters 5 and 10 (at least) of *geometry of physics – categories and toposes*, where the adjoint quadruples of the “standard models” for cohesion are constructed in detail as Quillen adjoint quadruples.

After that any number of examples/applications within these examples would fit well. If/since you have unpublished such examples worked out, that would be ideal to include, yes.

From my/our side there will probably also be a physics angle on these example, but just as motivation, not pushing itself into the foreground.

Right now we are in the last phase of finalizing another project. That should be out in week, I suppose. After that, if you are interested, I could start a shared Dropbox folder for a project *“Cohesive $\infty$-toposes for geometry and physics”*.

Re #85: How do you envision the content of such a paper? Is it going to consist of definitions, constructions, and computations? Or do you intend to include new difficult theorems?

Myself, I already wrote up quite a few examples (14 pages!) for my draft https://dmitripavlov.org/concordance.pdf, and I would certainly be interested in writing up more examples.

]]>Unfortunately this is not ready yet, at all.

I feel that there is a fair amount of worked examples scattered across dcct and articles like synthetic PDEs, but I gather these are unwieldy to extract for the reader. From my side, other projects that feel more urgent keep getting in the way.

But might you be interested in a collaborative effort?

]]>Yes, [SS20c] would be extremely helpful. If you have any preliminary drafts of [SS20c] (not necessarily complete), I could give them to the participants (not for redistribution, of course).

]]>Thanks for sending/forwarding all the feedback!

Yes, absolutely, I agree. That’s why there is a reference [SS20c] announced, meant to spell out examples. In a better world this would already be out.

]]>In Definition 3.25, the right diagram seemingly reverses the arrow f:Y→X to f:X→Y.

Here is some feedback from the seminar participants (not from myself, that is), feel free to ignore if it doesn’t seem apt.

For the quadruple adjunction in Definition 3.1 it would be very helpful to have the example of the site of cartesian manifolds presented in one place, i.e., give explicit formulas for all 4 functors in this special case. Right now there is only some scattered information in Examples 3.17, 3.18.

By the way, I recently realized that an easier way to see 3.18(iv) (but not the more general and powerful claim about the whole shape modality) is to observe that the inclusion Δ→Cart is an ∞-initial functor, so restricting along Δ^op→Cart^op preserves ∞-colimits.

It would also be very helpful to give at least one concrete nontrivial example for the induced 3 modalities and maps between them, e.g., what these 3 modalities do for the sheaf of principal G-bundles with connection.

For elastic toposes:

It would be very helpful if one could see concrete formulas for the simplest example, e.g., the inclusion of the site of cartesian manifolds into the Cahiers site.

Again, seeing the induced modalites evaluated on some nontrivial sheaf would go a long way.

Definition 3.29 of the tangent bundle appears rather unmotivated, if one is not familiar with the intuitive meaning of the de Rham stack functor ℑ. This also applies to 3.25.

The comments about elastic toposes also apply to solid toposes: again, having concrete formulas and an explicit example with computed modalities would help a lot.

(I supplied many examples to the participants myself, but I think many other readers of your paper will encounter similar issues.)

]]>Woops, that’s a couple of annoying typos. Thanks for catching! Have fixed it now in the pdf copy here.

]]>