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March 28, 2020

Pyknoticity versus Cohesiveness

Posted by David Corfield

Back to modal HoTT. If what was considered last time were all, one would wonder what the fuss was about. Now, there’s much that needs to be said about type dependency, types as propositions, sets, groupoids, and so on, but let me skip to the end of my book to mention modal types, and in particular the intriguing use of modalities to present spatial notions of cohesiveness. Cohesion is an idea, originally due to Lawvere, which sets out from an adjoint triple of modalities arising in turn from an adjoint quadruple between toposes of spaces and sets of the kind:

components \dashv discrete \dashv points \dashv codiscrete.

This has been generalised to the (,1)(\infty, 1)-categorical world by Urs and Mike. On top of the original triple of modalites, one can construct further triples first for differential cohesion and then also for supergeometry. With superspaces available in this synthetic fashion it is possible to think about Modern Physics formalized in Modal Homotopy Type Theory. This isn’t just an ‘in principle’ means of expression, but has been instrumental in guiding Urs’s construction with Hisham Sati of a formulation of M-theory – Hypothesis H. Surely it’s quite something that a foundational system could have provided guidance in this way, however the hypothesis turns out. Imagine other notable foundational systems being able to do any such thing.

Mathematics rather than physics is the subject of chapter 5 of my book, where I’m presenting cohesive HoTT as a means to gain some kind of conceptual traction over the vast terrain that is modern geometry. However I’m aware that there are some apparent limitations, problems with ‘pp-adic’ forms of cohesion, cohesion in algebraic geometry, and so on. In the briefest note (p. 158) I mention the closely related pyknotic and condensed approaches of, respectively, (Barwick and Haine) and (Clausen and Scholze). Since they provide a different category-theoretic perspective on space, I’d like to know more about what’s going on with these.

[Edited to correct the authors and spelling of name. Other edits in response to comments, as noted there.]

I’ll keep to the former approach since the authors are explicit in pointing out where their construction differs from the cohesive one. In cohesive situations, that functor which takes an object in a base category and equips it with the discrete topology has a left adjoint, and so preserves limits. This does not hold for pyknotic sets (BarHai 19, 2.2.4).

We hear

one of the main peculiarities of the theory of pyknotic structures … is also one of its advantages: the forgetful functor is not faithful. (BarHai 19, 0.2.4)

Where there is only one possible topology on a singleton set, in the category of pyknotic sets the point possesses many pyknotic structures.

The PykPyk construction applies to all finite-product categories, DD. There are several equivalent formulations of the concept, one being that Pyk(D)Pyk(D) is composed of the finite-product-preserving functors from the category of complete Boolean algebras to DD. (See others sites at pyknotic set.) We thus have Pyk(Ab)Pyk(Ab), the category of pyknotic abelian groups.

One reason, we are told, for the whole pyknotic approach is that Pyk(Ab)Pyk(Ab) rectifies a perceived problem with the category of topological abelian groups, AbTopAbTop, in that where the former is itself an abelian category, this is not the case with the latter:

This can be seen by taking an abelian group and imposing two topologies, one finer than the other. Both the kernel and cokernel of the continuous map which is the identity on elements are 0. This is an indication that AbTopAbTop does not have enough objects. To rectify this, we can modify the category to allow ‘pyknotic’ structures on 0, which can act as a cokernel here.

[Condensed abelian groups perform this role for Clausen and Scholze.]

PykPyk also preserves topos structure: If XX is a topos, then so is Pyk(X)Pyk(X).

I’m sure all the smart category theorists around here have useful things to say, but just to raise some small observations from cursory engagement.

Is there something importantly non-constructive about this construction? Complete Boolean algebras form the opposite category to Stonean locales, and

In the presence of the axiom of choice, the category of Stonean locales is equivalent to the category of Stonean spaces. (nLab: complete Boolean space).

Are there any category-theoretic features of CompBoolAlgCompBoolAlg being exploited, such as that it’s not cocomplete?

Concepts I’ve seen mentioned by Barwick include: ultraproduct, ultrafilter, codensity monad, proétaleness. There’s some connection between what they’re doing in BarHai19, sec 4.3 and Lurie’s work on Makkai’s conceptual completeness (see here), which Lurie is looking to extend to higher categories.

Barwick and Haines tell us of their Theorem 4.3.6 that

The main motivation of the study of 1-ultracategories is the following result, which implies both the Deligne Completeness Theorem and Makkai’s Strong Conceptual Completeness Theorem. (p. 35)

and refer to

  • Jacob Lurie, 2018, Ultracategories, (pdf).

This refers in turn to work by Scholze and Bhatt, presumably relating to why Scholze along with Clausen have devised a close relative of pyknoticity in condensed mathematics. A condensed set is a sheaf of sets on the pro-étale site of a point.

Awodey and students (Forssell and Breiner) were looking for an alternative route to this model-theoretic area (avoiding ultra-structures in favour of topological ones, see pp. 6-7 of Forssell, and even of scheme-theoretic ones in Breiner):

we reframe Makkai & Reyes’ conceptual completeness theorem as a theorem about schemes. (Breiner, p. 9)

Seems an interesting tangle of ideas.

Posted at March 28, 2020 6:02 PM UTC

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Re: Pyknoticity versus Cohesiveness

I think Clausen deserves mentioning in the “Condensed” camp, and from what I’ve heard/read the ideas originated (independently) with him some years back.

Posted by: David Roberts on March 28, 2020 11:02 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Yes, both sets of co-authors should be corrected.

The pyknotic team is Clark Barwick and Peter Haine.

The condensed team is Dustin Clausen and Peter Scholze. There’s some relation with the Bhatt-Scholze work on the pro-é topology for schemes, but the condensed mathematics framework is described as joint work of Clausen and Scholze.

David, if it would be helpful if I went into the back end of the café to correct this in the blog post, I’d be happy to.

Posted by: Emily Riehl on March 29, 2020 2:24 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Along the same lines as Emily Riehl’s comment, one should remark that the condensed sets/abelian groups/etc already show up in Bhatt-Scholze (under a different name), as does the the important idea of using Stone-Cech compactifications to obtain a nice basis for the topos. The new contribution of Clausen-Scholze is the introduction of the idea of solidity (and later liquidity) and its many amazing properties.

Posted by: A different ER on March 29, 2020 4:50 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks, I added a note to my post on your first point. Do these other good features – solidity and liquidity – carry over to pyknotic sets?

I see Barwick and Haine write

As emphasised by Scholze, however, the distinction between pyknotic and condensed does have some consequences beyond philosophical matters. For example, the indiscrete topological space {0,1}\{0,1\}, viewed as a sheaf on the site of compacta, is pyknotic but not condensed (relative to any universe). By allowing the presence of such pathological objects into the category of pyknotic sets, we guarantee that it is a topos, which is not true for the category of condensed sets.

It would be too glib to assert that the pyknotic approach values the niceness of the category over the niceness of its objects, while the condensed approach does the opposite. However, it seems that the pyknotic objects that one will encounter in serious applications will usually be condensed, and the majority of the good properties of the category of condensed objects will usually be inherited from the category of pyknotic objects. (BarHai 19, 0.3)

Posted by: David Corfield on March 30, 2020 8:22 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks both for pointing out my slips. I’ve corrected them now.

Posted by: David Corfield on March 29, 2020 10:35 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

What applications of cohesive HoTT and pyknotic or condensed sets are there to quantum chemistry and condensed matter physics?

Posted by: Madeleine Birchfield on March 29, 2020 8:25 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

No idea for pyknotic or condensed sets. To answer for cohesive HoTT, I’d point towards the kind of lines of research in

  • Daniel Freed, Gregory Moore, Twisted equivariant matter, Ann. Henri Poincaré (2013) 14: 1927 (arXiv:1208.5055)

Twisted equivariant K-theory is just the terrain for cohesive HoTT as a twisted cohomology and an equivariant cohomology.

Another place to point would be AdS-CFT in condensed matter physics.

Posted by: David Corfield on March 29, 2020 10:56 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Non-faithfulness of the forgetful functor is a separate issue from non-existence of a left adjoint. In fact the forgetful functor for cohesive toposes is also almost never faithful; this is basically the price you pay for having a topos (or topos-like category) of spaces rather than a quasitopos (or quasitopos-like category). And as I’ve pointed out elsewhere, non-faithfulness is also what rectifies the problem of quotients. So none of that explains why the pyknotic world diverges from the cohesive one.

Posted by: Mike Shulman on March 29, 2020 11:27 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks. That was misleading on my part to give the impression by running on from the lack of left adjoint to the non-faithfulness of the underlying functor that it was part of the same account of the difference from cohesion. So I’ve set it as a new paragraph.

Posted by: David Corfield on March 30, 2020 8:34 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

I can try to say something about what I hope may be true of the pro-étale (,1)(\infty, 1)-topos. However, I am not an expert in any sense, and have not studied the pro-étale site in detail at all. I would be delighted if an expert can along and confirm or deny some of what I write below.

A picture I am very fond of, which goes back essentially to Grothendieck’s letters to Breen in the mid-70s in which the homotopy hypothesis was introduced, is that cohomology of some object XX should be viewed as coming from Hom(π (X),K(A,n))Hom( \pi_{\infty}(X), K(A, n)), where π (X)\pi_{\infty}(X) is the ‘fundamental \infty-groupoid’ of XX, where this HomHom is the HomHom of an (,1)(\infty, 1)-topos, and where K(A,n)K(A,n) is the nn-th Eilenberg-MacLane space of AA viewed as a constant object of this (,1)(\infty, 1)-topos.

That this picture more or less holds goes back to Artin and Mazur’s construction of the étale homotopy type of a scheme, and this work was heavily influential both on Grothendieck’s letters to Breen and on Pursuing Stacks. There has been much later work re-incarnating Artin and Mazur’s work in different foundational settings; one of the most important is Toën’s work, partly with Vezzozi, in the early 2000s, which was the first to do it, very conceptually, in an (,1)(\infty, 1)-categorical setting.

The key words in the previous paragraph are, however, ‘more or less’. Even in Toën and Vezzosi’s work as well as in more recent variants on that, including quite recent work of some authors working with Barwick, it is not exactly true in general that π (X)\pi_{\infty}(X) is an object in the same (,1)(\infty, 1)-topos as K(A,n)K(A,n). For this to be the case, the XX which one begins with needs to live in a ‘locally contractible (,1)(\infty,1)-topos’, i.e. an (,1)(\infty,1)-topos TT for which the canonical (,1)(\infty, 1)-functor from \infty-groupoids to TT sending an \infty-groupoid to the ‘locally constant sheaf on it’ admits a left adjoint. This left adjoint is exactly π \pi_{\infty} in such cases, and is given explicitly by Artin and Mazur’s construction.

In general, π (X)\pi_{\infty}(X) is some other kind of gadget, some kind of pro-\infty-categorical gadget. In particular, this is the case in algebraic geometry with the étale (,1)(\infty,1)-topos of a scheme.

What excites me about Bhatt and Scholze’s work is that there are some results in it which seem to indicate to me that in the big pro-étale (,1)(\infty,1)-topos TT over some appropriate field kk, one might be able make sense of π (X)\pi_{\infty}(X) inside TT. It might even be that TT is locally contractible. Bhatt and Scholze introduce the notion of ‘w-contractibility’, but the results on locally constant sheaves which they obtain suggest that TT might actually really be locally contractible (all this in the (,1)(\infty,1)-topos theoretical sense), or at least that π (X)\pi_{\infty}(X) exists with the correct universal property under mild conditions on XX.

As I say, I am no expert, and this is more of a hope than a knowledgeable assertion. But I think the picture cannot be far off.

Why is this important? Just as Bhatt and Scholze discuss in their paper, this may seem a very technical point at first, but it makes it possible to apply general techniques around cohomology inside (,1)(\infty, 1)-toposes that otherwise are forbiddingly difficult to replicate in ad-hoc ways.

I can add a couple of further remarks. Firstly, to give a bit of context, I contributed the locally contractibility part of the notion of a cohesive (,1)(\infty, 1)-topos to Urs over a decade ago now(!), or at least suggested some ideas to him that aligned with other hints he had at the time. I always felt that this part was much more significant than the additional structure around discreteness and co-discreteness. In particular, I would not be worried about condensed or pyknotic sets not being discrete, there is still much one can do if one has local contractibility.

Secondly, my own motivation back at that time was actually in algebraic geometry, more specifically motivic homotopy theory. What I wanted to do was to make the above picture work for motivic cohomology. There are three main obstacles to this. The first one is possibly cleared up by Bhatt and Scholze’s work, namely finding an appropriate (,1)(\infty, 1)-topos which is locally contractible or something close.

The second is the role of Tate twists. I have always felt this to be the most important aspect. Even for ordinary étale homotopy theory (ignoring motivic homotopy theory for a moment), one can ask: can modify the above picture to bring in Tate twists? I.e. replace K(A,n)K(A, n) by some other gadget K(A,n)(i)K(A,n)(i) which recovers H n(X,A(i))H^{n}(X, A(i))? It is very important that the replacement of K(A,n)K(A,n) should be a sheaf-level operation, i.e. we are not interested in the construction K(A(i),n)K(A(i), n). Now, one could hope that this Tate twist operation is simply a smash product of pointed constant objects in an (,1(\infty, 1)-topos, namely something like K(A,n)𝔾 mK(A,n) \wedge \mathbb{G}_{m} (where both are pointed so that this makes sense).

The third is the role of 𝔸 1\mathbb{A}^{1}-localisation. It is well known that this localisation is not left exact, i.e. one does not obtain an (,1)(\infty, 1)-topos if one localises a big étale (,1)(\infty, 1)-topos in this way. However, Bhatt and Scholze’s work, if it can be re-interpreted (,1)(\infty, 1)-topos theoretically as above, offers the intriguing possibility of considering not 𝔸 1\mathbb{A}^{1} but π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1}). The latter gadget in the pro-étale case, being related directly to l\mathbb{Q}_{l}, is more like a characteristic 0 object, and the affine line has contractible étale homotopy type (in the usual sense) in characteristic zero. It might be that it is not even necessary to do any kind of 𝔸 1\mathbb{A}^{1}-localisation if one works with π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1}). Or at very least, since π ()\pi_{\infty}(-) preserves colimits, and since all schemes are colimits of affine ones, there is a good chance I think that the localisation will be left exact (one might restrict to gadgets of the form π (X)\pi_{\infty}(X) rather than all XX). This is pure speculation, but it is an idea.

What would be the good of overcoming these three obstacles? The dream was to give a conceptual proof of the Beilinson-Soulé conjecture, probably the most important open problem in motivic cohomology, which states that ‘motivic cohomology vanishes in negative degree’. If one is able to view motivic cohomology Hom(π (X),K(A,i)(j))Hom(\pi_{\infty}(X), K(A,i)(j)) in some (,1)(\infty,1)-topos, recalling that K(A,i)(j)K(A,i)(j) is some (,1)(\infty,1)-theoretic version of a Tate twist, i.e. smashing with the multiplicative group, then it is a triviality that it vanishes in negative degree. I think that a comparison theorem with ordinary motivic cohomology would not be too bad to obtain, it would be very similar to the comparison theorems in the Bhatt-Scholze paper I think.

Posted by: Richard Williamson on March 30, 2020 1:05 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

What seems an important aspect of cohesion is that it allows for the amalgamation of topological cohomology classes with curvature differential forms, (see nLab: differential cohomology diagram).

Note the use there of the homotopy fiber and cofiber of (the suspension/looping of) the unit and counit associated to the two modalities. So here for spectra at least we don’t see that problem of vanishing (co)kernels?

Posted by: David Corfield on March 30, 2020 11:35 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

As I said in my previous comment, the problem of (co)kernels is also solved by cohesion, in the same way that it is solved by pyknoticity: the forgetful functor is not faithful.

Posted by: Mike Shulman on March 31, 2020 2:16 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Related to the question of Williamson, let me make some remarks that seem relevant to the discussion.

Caveat 1: I do not really know or understand cohesiveness, so what I am saying may be off.

Caveat 2: I regard the distinction between condensed and pyknotic things as very small. For me, pyknotic sets are κ\kappa-condensed sets for some suitable chosen cutoff cardinal κ\kappa, and condensed sets are the union over all possible choices of κ\kappa. This retains all properties of a topos, except that one has not a set, but a class of generators. In practice, this makes little difference, as in practice one can deduce anything one wants by taking a colimit over all choices of κ\kappa. A few things that exist in pyknotic condensed sets but not in condensed abelian groups (e.g., injective pyknotic abelian groups) turn out in practice to be of very little use from my experience. Whenever I tried to use them, the argument turned out to have an error. In the following, I mostly ignore the distinction between condensed and pyknotic.

From what I understand, cohesiveness can be understood as a relative notion for a map of topoi f:YXf: Y\rightarrow X; usually, one always chooses XX to be the punctual topos, i.e. the category of sets. If I understand it right, this would mean that in addition to the functors f *f^\ast and f *f_\ast, there is an additional right adjoint of f *f_\ast and an additional left adjoint of f *f^\ast.

As mentioned above, this fails for the map f:* proet* etf: \ast_{\mathrm{proet}}\rightarrow \ast_{\mathrm{et}} from the pro-etale topos of a point to the etale topos of a point (=the punctual topos), as there is no left adjoint of f *f^\ast. (The existence of the right adjoint is a matter of condensed vs. pyknotic: It exists only in the pyknotic case. I have never seen a good use of it.)

However, here is some nontrivial result that gives a concrete incarnation of the assertion that the pro-etale topos of a scheme is “locally contractible” and seems to be in the spirit of cohesiveness.

Namely, for any scheme XX, consider the map f:X proet* proetf: X_{\mathrm{proet}}\rightarrow \ast_{\mathrm{proet}} from the pro-etale topos of XX to the pro-etale topos of a point.

Proposition. f *f^\ast admits a left adjoint.

The proposition is true both on the level of usual topoi, and on the level of \infty-topoi of hypercomplete sheaves of anima (sorry ;-) ).

In particular, applying this left adjoint to XX itself, in the \infty-topoi setting, produces a sheaf of anima on * proet\ast_{\mathrm{proet}}, i.e. a condensed anima. This is the π (X)\pi_\infty(X) that Williamson wants, and is some incarnation of the etale homotopy type of XX.

In fact, the preceding Proposition applies for any map f:YXf: Y\to X of schemes.

So, when one starts doing condensed(/pyknotic) mathematics, one should at the very beginning replace the topos of sets (or the \infty-topos of anima) by the (macro)topos of condensed sets (or the \infty-(macro)topos of condensed anima), and accordingly instead of asking cohesiveness relative to sets, one should ask for cohesiveness relative to condensed sets, and this is what’s true for the pro-etale topos of any scheme.

Well, except that f *f_\ast does not usually have a right adjoint! (As I don’t know anything about cohesion, I can’t comment on the relevance of that right adjoint. It does not seem related to the question of local contractibility.) If I had to guess, the only case where it has one is the (trivial) case where XX is a finite disjoint union of geometric points. The non-existence of this right adjoint here is not a matter of condensed vs. pyknotic.

Posted by: Peter Scholze on March 30, 2020 11:40 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

The additional right adjoint in cohesion gives rise to the sharp modality which is used to talk about concrete objects and concretification.

Also, it plays a role in cohesive HoTT:

The crucial insight of Mike Shulman [Shu11] is that to implement cohesion fully formally in homotopy-type theory one is to regard the sharp modality \sharp, remark 4.1.12, as the fundamental axiom that serves to exhibit the external base ∞-topos as an internal sub-system of homotopy-types. Then the flat modality and the shape modality are axiomatized based on the existence of the sharp modality. (Differential cohomology in a cohesive ∞-topos v.2, p. 314)

Posted by: David Corfield on March 30, 2020 12:40 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

It should be noted, however, that in Mike’s Real Cohesion, the further right adjoint modality \sharp is not used in the definition of the middle modality \flat, nor in the definition of the left modality π \pi_{\infty}. (It is used to prove some facts about \flat.)

So long as the map ff from the pro-etale site of XX to the pro-etale site of the point point has a fully faithful inverse image f *f^{\ast}, I believe that the \flat portion of Mike’s Cohesive Type Theory could be used.

Posted by: David Jaz Myers on March 30, 2020 10:08 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

OK, I realize there’s more to cohesion. Can anybody tell me the axioms?

The functor f *f^\ast will generally be far from fully faithful. If I’m not screwing up, it is fully faithful precisely when XX is strictly henselian.

So the only thing I’m claiming in general is the existence of that left adjoint to f *f^\ast, and that this is a manifestation of some form of “local contractibility” of the pro-etale site.

Posted by: Peter Scholze on March 30, 2020 11:09 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

The definition of a cohesive \infty-topos is here.

Posted by: David Corfield on March 31, 2020 7:37 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thank you very much Professor Scholze for joining in the discussion, it is greatly appreciated. This is fantastic, it is exactly what I was looking for indeed!

I think this formalism will prove over time to be a big technical breakthrough. I hope that others more knowledgeable than me can join in the discussion here.

I can ask a possibly naïve follow-up question myself. It would be very beautiful if in the picture of cohomology which I outlined, we could recover with the pro-étale π (X)\pi_{\infty}(X) not just l-adic cohomology as an abelian group but as a Galois representation, i.e. capture in some conceptual/canonical way the arithemetic aspect. I can imagine that using pro-étale topos of the point as the base as you describe would facilitate this. Are you able to make that somewhat precise in some way already? More specifically:

1) I imagine that there is some condensed \infty-groupoid (animum?) version of an Eilenberg-MacLane space. Do you expect to be able to recover l-adic cohomology by mapping from π (X)\pi_{\infty}(X) into these Eilenberg-MacLane space analogues?

2) Presumably π (Spec(k))\pi_{\infty}(Spec(k)) is something closely related to Galois theory, some kind of ‘Galois \infty-groupoid’? Any idea what the extra information in this as opposed to the plain Galois group can be?

3) If 1) and 2) are true, do we perhaps get some kind of canonical ‘Galois action’ on π (X)\pi_{\infty}(X) itself using π (Speck)\pi_{\infty}(Spec k)?

4) If 3) is true or if it is not, can we use 1) and 2) to recover in some canonical way l-adic cohomology as a Galois representation using Hom(π (X),K(A,n))Hom(\pi_{\infty}(X), K(A,n))?

5) Would Tate twists be able to be seen as coming from π ( 1)\pi_{\infty}(\mathbb{P}^{1}) in some way?

If all of 1) - 5) are completely off the mark, is there a different way that you see to get the Galois action?

I realise that these questions may be reaching too far into the future, but I would be delighted to hear any thoughts that you would have, or any other passing expert would have, around this! Thanks again for what you have already contributed to the discussion.

Posted by: Richard Williamson on March 30, 2020 3:23 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

I think much of this can be done, already from what I said in the previous message.

(Regarding terminology: Clausen and I call \infty-groupoids “anima” (plural: anima). We regard the name \infty-groupoid as too unwieldy – the name suggests a semblance to groups, which is not really a good first approximation. Lurie’s choice of “spaces” does not work in the setting of condensed mathematics: Condensed anima have two directions, a homotopy-theoretic direction (anima) and an actual topological space direction (condensed), and it is the condensed direction that feels like the space direction, so the term condensed spaces would be a complete misnomer. On the other hand, see my paper with Cesnavicius (on purity for flat cohomology) for some discussion of anima and animation, including some justification for that name.)

1) Yes. This follows formally from adjointness.

2) Yes, it is the condensed anima BGBG where G=Gal(k¯/k)G=\mathrm{Gal}(\overline{k}/k) is the absolute Galois group of kk. There is no extra information.

3) If XX is a scheme over a field kk, you can consider f:X proetSpeck proetf: X_{\mathrm{proet}}\rightarrow \mathrm{Spec}\, k_{\mathrm{proet}}. Sheaves on Speck proet\mathrm{Spec}\, k_{\mathrm{proet}} are equivalently condensed anima with GG-action (where GG is the absolute Galois group as above), by descent. Applying the left adjoint to XX then produces a GG-action on π (X k¯)\pi_\infty(X_{\overline{k}}).

4) Yes, you can take that Hom internally on Speck proet\mathrm{Spec}\, k_{\mathrm{proet}}.

5) I presume one can get \ell-adic Tate twists (but not motivic Tate twists!) in some way out of π ( 1)\pi_\infty(\mathbb{P}^1).

However, I should also say that I am rather sure none of this has any relevance to Beilinson-Soule.

Posted by: Peter Scholze on March 30, 2020 4:18 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

PS: 1) I should have said that the name “anima” is heavily inspired by Beilinson’s paper on topological ϵ\epsilon-factors.

2) With / n\mathbb{Z}/\ell^n\mathbb{Z}-coefficients, the relevant Eilenberg-MacLane “spaces” are just the usual Eilenberg-MacLane anima, noting that anima embed fully faithfully into condensed anima, as the objects that are discrete in the condensed direction. When working with \mathbb{Z}_\ell-coefficients, one simply takes the limit of the objects for / n\mathbb{Z}/\ell^n\mathbb{Z}-coefficients. Alternatively, for any condensed abelian group AA and n1n\geq 1, one can form K(A,n)K(A,n) as a (pointed) condensed anima – this works on any topos. Working with Speck proet\mathrm{Spec}\, k_{\mathrm{proet}}, one can also apply this to the usual \ell-adic Tate twists and get K( (i),n)K(\mathbb{Z}_\ell(i),n).

Posted by: Peter Scholze on March 30, 2020 4:29 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

OK, here is another try to get something that looks closer to cohesion.

For a scheme XX, one can consider the small pro-etale topos X proetX_{\mathrm{proet}} built from the site of schemes pro-etale over XX, and pro-etale coverings, and the big pro-etale topos X ProetX_{\mathrm{Proet}} built from the site of all schemes over XX, and pro-etale coverings. There is a natural map of topoi f:X ProetX proetf: X_{\mathrm{Proet}}\to X_{\mathrm{proet}}.

Proposition. f *f^\ast has a left adjoint, and in the pyknotic formalization, f *f_\ast has a right adjoint. Moreover, f *f^\ast is fully faithful.

Taking XX to be a geometric point Speck\mathrm{Spec}\, k, the target X proetX_{\mathrm{proet}} here is the topos of pyknotic sets. (In the condensed formalization, I’m afraid the right adjoint may fail to exist. Again, I’m a little suspicious about its use. Concreteness also does not seem like a fundamental concept to me.)

Again, all assertions are true both on the level of usual topoi, and on \infty-topoi of hypercomplete sheaves.

However, there is still one bit missing: That the left adjoint commutes with finite products. This is actually a nontrivial question, nontrivial approximations to which are true (related to Künneth formulas in etale cohomology) but I currently doubt it is true in this form. It comes down to the following:

Question. Let CABC\leftarrow A\to B be a diagram of strictly henselian local rings and local maps, and let X=Spec(C AB)X=\mathrm{Spec}\, (C\otimes_A B). Is π (X)\pi_\infty(X) equivalent to a point?

Posted by: Peter Scholze on March 31, 2020 8:21 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Confirmed my suspicion: This is not true.

A related fact is that if XX is a scheme over an algebraically closed field kk, and k/kk'/k is an extension of algebraically closed fields, with XX' the base change of XX to kk', then π (X)π (X)\pi_\infty(X')\to \pi_\infty(X) is not an isomorphism, in general.

Some approximations to such statements are true, for example invariance of \ell-adic cohomology under change of algebraically closed base field, if \ell is different from the characteristic of kk. But, for example, this fails when \ell equals the characteristic of kk.

Posted by: Peter Scholze on March 31, 2020 1:49 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Regarding concreteness: Let’s take X=SpeckX=\mathrm{Spec}\, k with kk algebraically closed as our base. There are many schemes YY over kk with points Y(k)Y(k) empty: For example, Y=SpeckY=\mathrm{Spec}\, k' for some algebraically closed field k/kk'/k, kkk'\neq k. In particular, this implies that any concrete sheaf FF satisfies F(Y)=*F(Y)=\ast for all such YY. But such FF are really awkward – in practice F(k)F(k) will always inject into F(k)F(k'). Not even the disjoint union of two points is concrete!!

So in this setting concrete sheaves are a strange concept. In algebraic geometry, schemes are not built from kk-valued points.

Posted by: Peter Scholze on March 31, 2020 2:03 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Concerning local contractibility, Barwick and Haine’s exodromy paper shows that, for a scheme XX, the pro-étale topos of XX is not only locally contractible over the pro-étale site of the point: it is an internal presheaf topos. About concreteness, we could also focus on arithmetic schemes (e.g. schemes of finite type over Z\mathbf Z or over a number field). For instance, we know from Voevodsky (reinterpreted by Barwick and Haine) that assigning the (pro-)étale topos to a scheme of finite type over a number field defines a fully faithful functor. Wouldn’t that be relevant?

Posted by: Denis-Charles Cisinski on March 31, 2020 2:45 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks for this!

How is that result of Voevodsky related to concreteness? The phenomenon I mention seems to exist also in arithmetic settings; and with any scheme you are also forced to include all schemes pro-etale over it, so you also need to include further algebraically closed fields. But I’m probably misunderstanding what you are hinting at.

Posted by: Peter Schlze on March 31, 2020 3:14 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Well, I do not know very much about cohesive topoi or concreteness. But I was reacting to your comments about schemes not being built from kk-valued points. Voevodsky’s embedding result together with Barwick and Haine’s exodromy say that one can reconstruct an arithmetic scheme of characteristic zero from the condensed/pyknotic category of its points. In fact, I just had a look before posting this, so here is the exact hypothesis (it looks like I was a little but too optimistic about the level of generality for our schemes): we have to work with normal kk-schemes of finite type with kk a field of characteristic zero of finite type, and the fully faithfulness is not relatively to the pro-étale site of the point, but the one of Spec(k)Spec(k). See Theorem 14.4.7 in the Exodromy paper. This means that we can recontruct a normal kk-scheme of finite type from its KK-points where KK runs over all extensions of finite type K/kK/k, at least in some sense.

Posted by: Denis-Charles Cisinski on March 31, 2020 10:17 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Ah, OK, thanks for the clarification!

For all I can see, this is unrelated to the question of concreteness in the context of cohesion (in the context of the example I considered) – the latter takes into account only X(k)X(k) for the fixed base field kk. I tend to agree that asking for concreteness in that example is probably not an interesting question to ask, but simply wanted to make some concrete (ha) statement justifying this.

Posted by: Peter Scholze on March 31, 2020 10:38 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Dear Denis-Charles, thank you for joining the discussion! You probably do not remember, but we once had a walk around the University Parks on a very pleasant day in Oxford, it must be about 10-11 years ago, I think in the spring, discussing things not too far from what is being discussed here! Raphaël told me afterwards that you said to him something like that what we were discussing was very interesting, but that it would take at least 15 years to do!!!

Anyhow, to get back on topic, I could not immediately see how the exodromy paper that you refer to produces a π \pi_{\infty} functor/exhibits local contractibility. The gadget Π (,1) et,(X)\Pi^{et, \wedge}_{(\infty,1)}(X) in the exodromy paper is a pro-object, whereas the key aspect of π (X)\pi_{\infty}(X) is that it is an actual condensed \infty-groupoid/anima. Of course condensed anima are themselves closely related to pro-objects, but still I think the distinction is fundamental to the discussion here.

Scholze would of course know infinitely better than me, but his construction of π (X)\pi_{\infty}(X) will I think, like the construction of the fundamental group in the paper with Bhatt, follow a ‘classical’ pattern using locally constant stacks in the pro-étale topology. The hard work is in establishing the necessary properties of locally constant stacks in this topology; I imagine that the (,1)(\infty,1)-version of this is not much different from the work in the Bhatt-Scholze paper. From this, I think one will be to construct the adjunction in à la Toën-Vezzosi and your own work on this.

What I am trying to say is that, just given the bare definition of the pro-étale topology and the exodromy paper, it does not seem straightforward to me to say anything about local contractibility; one has to do some hard work in the pro-étale topos. After the fact, maybe it is possible to make some identification between π (X)\pi_{\infty}(X) and the objects in the exodromy paper; is this the kind of thing you meant?

Posted by: Richard Williamson on March 31, 2020 11:28 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

The Exodromy construction follows a very classical pattern as well: instead of constructible locally constant sheaves, we consider all constructible sheaves. Pro-finite categories may be seen as categories internally in the topos of sheaves on the site of Stone spaces (the topology is created by the limit of discrete finite categories), which we may also interpret as topological categories. Therefore, what Exodromy provides is a category in the pro-étale topos of the point. They associate to each topos such a topological category. This induces a full embedding of a 22-category of the one of topoi, the objects of which are called spectral, into the 22-category of condensed/pyknotic categories. An example of spectral topos is the small étale topos of a scheme. Therefore, one can reconstruct the small étale topos of a scheme XX from its Galois category Gal(X)Gal(X). If XX is a scheme, the pro-étale topos of XX can be recovered from Gal(X)Gal(X) easily: we take presheaves on Gal(X)Gal(X) internally in the pro-étale topos of the point (i.e. “presheaves in the condensed/pyknotik sense”), which thus gives an internal topos, whence an internal category, the global sections of which is the pro-étale topos of XX. In other words, the internal site corresponding to X proetX_{proet} is Gal(X)Gal(X) with the (internal) coarse Grothendieck topology. This is why X proetX_{proet} is “internally locally contractible”, as is any presheaf category: the condensed/pyknotik pro-homotopy type of XX is defined by inverting all maps in Gal(X)Gal(X) to obtain a univalent \infty-groupoid within condensed/pyknotik anima, hence a condensed/pyknotik anima.

Posted by: Denis-Charles Cisinski on April 1, 2020 3:01 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Actually, where in the exodromy paper of Barwick and Haine do they state this result that the pro-etale topos of XX is an internal presheaf topos over pyknotic sets? For most of that paper, they seem to work with pro-objects, which, as Williamson observes, is close but also quite different.

Posted by: Peter Scholze on April 1, 2020 10:09 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

They do not prove that in the Exodromy paper: this will appear in a sequel and is announced explicitely in the introduction of their paper Pyknotic objects, I. Basic notions (page 2). (About pro-finite categories vs condensed/pyknotik categories: I just thought that if we have a site with finite limits, it is obvious that an internal category of the site defines an internal category of the corresponding topos via Yoneda, and thus a sheaf of categories.)

Posted by: Denis-Charles Cisinski on April 1, 2020 3:12 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks for the reference! I had seen that assertion but hadn’t made that connection.

Their Gal(X)\mathrm{Gal}(X) indeed easily transfers from pro-object to pyknotic/condensed category, as everything is somehow profinite. But if you want π (X)\pi_\infty(X), you need to invert all arrows, and then the order matters.

Posted by: Peter Scholze on April 1, 2020 9:47 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

As has been said, not all of cohesion is needed to generate interesting consequences. One important part of the story is how the differential cohomology diagram provides an account of differential cohomology. This doesn’t need the right adjoint to f *f_{\ast}. Urs suggests that f !f_! doesn’t need to preserve finite products either, so generalized Chern characters, etc., should follow in this pro-étale situation.

Posted by: David Corfield on April 1, 2020 11:09 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Cohesiveness is intended as a formalization of a “big topos”, so I would never have expected it to hold for things like X proetX_{proet} which are “little toposes”. Sheaves on some fixed topological space are also not cohesive; it’s sheaves on some fixed category of contractible test spaces (like Euclidean spaces) that is cohesive.

Posted by: Mike Shulman on March 31, 2020 5:35 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Also, I would not over-emphasize concreteness. It’s true that \sharp can be used to formalize concreteness, but one of the whole points of cohesion is that not all interesting objects are concrete (in fact, non-concreteness is another way of describing the solution to the problem of (co)kernels), so it’s unclear that this is of much real value. As David Jaz Myers pointed out, \flat and ʃʃ are of at least as much real use in doing mathematics cohesively. The lack of ʃʃ is, I think, what leads to a significantly different flavor more than the lack of \sharp.

Posted by: Mike Shulman on March 31, 2020 5:59 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

\sharp is having a hard time on this thread, but in case anyone hasn’t seen this, Lawvere’s account of cohesion involved the idea of overcoming a fundamental opposition between the idempotent comonad constant on the initial object and the idempotent comonad constant on the terminal object, sketched here.

Posted by: David Corfield on March 31, 2020 6:32 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks, this is very helpful! As I said, I was/am struggling to see why a cohesive topos is defined like it is, but I am beginning to see some light.

I guess the above example, that the big pro-etale topos of schemes over a fixed algebraically closed field kk is, almost, cohesive over pyknotic sets, is fitting this philosophical framework then?

Posted by: Peter Scholze on March 31, 2020 8:01 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

I guess the above example, that the big pro-etale topos of schemes over a fixed algebraically closed field kk is, almost, cohesive over pyknotic sets, is fitting this philosophical framework then?

Yes.

Posted by: Mike Shulman on April 1, 2020 1:05 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks very much again, Professor Scholze, for all the contributions here. Thank you in particular for confirming my suggestions around the Galois action. Though these are more or less formalities, I feel this is kind of the point; that this technical innovation makes all sorts of quite important things completely formal.

There are many more things which I think follow completely formally from the existence of π (X)\pi_{\infty}(X). For example, all ‘ordinary’ spectra (without any condensed aspect) should give rise to a condensed spectrum (any terminology for these :-)?), and hence some kind of generalised ll-adic cohomology theory for XX. The Eilenberg-MacLane spectrum will give back ll-adic cohomology. But we also can just plug in the complex topological KK-theory spectrum BUBU. This is some version of étale KK-theory of XX I think. We should be able to get some results relating ll-adic cohomology and this flavour of étale KK-theory more or less for free I think. Similarly for elliptic cohomology, cobordism, Morava K-theories, etc, etc. All of these should have Galois actions as well coming from that on π (X)\pi_{\infty}(X). A lot of results about stable homotopy theory probably translate more or less immediately to these ll-adic versions. Things like the construction of Chern classes for example can probably be made completely formal. But one will be able to use the geometry and arithmetic of XX as well, of course, to study these ll-adic theories. We can note that XX is quite general here, as well.

On similar lines, one should be able to construct a stable (,1)(\infty,1)-category whose homotopy category is an unbounded derived category of l\mathbb{Q}_{l}-sheaves on XX by looking at (,1)(\infty,1)-functors from π *(X)\pi_{*}(X) into H lModH\mathbb{Q}_{l}-\mathsf{Mod}, where H lH\mathbb{Q}_{l} is the (condensed version of) the Eilenberg-MacLane spectrum for l\mathbb{Q}_{l}, and H lModH\mathbb{Q}_{l}-\mathsf{Mod} is the category of condensed module spectra associated to it.

Do you agree?

On a different note, I completely agree that there is nothing motivic about the constructions which we have discussed so far. But I am optimistic that this picture can translate to the motivic world (we can forget about Beilinson-Soulé, this was just an old dream that I probably should not have mentioned, even if I have not entirely given up on it!).

For instance, I think there should be a Nisnevich version of the pro-étale topology. Let us say pro-Nisnevich, even if this sounds rather political! The most naïve guess would be just to say that a morphism of schemes is weakly Nisnevich if instead of requiring ‘étale + …’ as for Nisnevich morphisms we just require ‘weakly étale + …’. And then define the pro-Nisnevich site using pro-Nisnevich morphisms and fpqc covers. I would imagine that this works out perfectly fine, and that one can construct a perfectly good π (X)\pi_{\infty}(X) in the Nisnevich setting too. What do you think?

And then we might consider my earlier suggestion of using not 𝔸 1\mathbb{A}^{1} but π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1}) to define 𝔸 1\mathbb{A}^{1}-localisation. I.e. the localisation could be not of the Nisnevich (,1)(\infty,1)-topos of schemes but of condensed anima (or maybe the big pro-étale topos over Spec(k)Spec(k)) at morphisms π (X)×π (𝔸 1)π (X)\pi_{\infty}(X) \times \pi_{\infty}(\mathbb{A}^{1}) \rightarrow \pi_{\infty}(X). It is an interesting question whether this localisation is left exact, i.e. whether one gets an (,1)(\infty,1)-topos. I think there is a chance one does. I think it may be related to Kunneth theorems in étale cohomology where one of the schemes involved is 𝔸 1\mathbb{A}^{1}, to touch on something similar to one of your other comments.

I mentioned earlier the question of whether it is necessary to localise at π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1}) at all, i.e. whether it might already be contractible, but I suppose on further reflection that this would be too good to be true.

Even if localising using π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1}) does not give a topos, one can still construct HomHom into an Eilenberg-MacLane space in the resulting (,1)(\infty,1)-category. This must be something like algebraic singular cohomology, surely? And then one can certainly try smashing the Eilenberg-MacLane space with something like π ( 1)\pi_{\infty}(\mathbb{P}^{1}) or maybe something involving 𝔾 m\mathbb{G}_{m} to get something like motivic Tate twists.

What do you think?

As a final PS, let me say that I’ll leave it to others to discuss your interesting remarks about the right adjoint part of cohesion and concreteness more. I can just say, as I hinted in my original comment, that I agree with you that I not fully convinced about the importance of it conceptually; at least it is certainly not as important in my opinion as π \pi_{\infty}.

Posted by: Richard Williamson on March 31, 2020 3:34 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

There’s a lot of questions here, and I’m not sure I have something sensible to say about many of them.

Regarding π (X)\pi_\infty(X) versus the derived category of constructible sheaves: Using π (X)\pi_\infty(X) you can only see the locally constant constructible sheaves. If you want to see all constructible sheaves, you should use Barwick-Haine’s Gal(X)\mathrm{Gal}(X), a condensed/pyknotic category, and then this is their exodromy theorem.

You can definitely build a pro-Nisnevich topology. I don’t see its relevance – I am one of the few people who have never understood why one wants to consider the Nisnevich topology when one is studying motives. Note that rationally, everything is unchanged by working with the etale topology, and all interesting questions about motives (in the original sense) are rational (like Beilinson-Soule, etc.etc.).

But yes, you should be able to proceed, and build a Nisnevich version of π (X)\pi_\infty(X). This would however know only about the Nisnevich cohomology of constant sheaves, and all interesting Nisnevich sheaves are not constant.

I don’t think the Künneth formula will be true if one of the factors is 𝔸 1\mathbb{A}^1 in either the pro-Nisnevich or the pro-etale settings; in the pro-etale setting, the Artin-Schreier problem in characteristic pp persists.

And I think you can’t possibly get algebraic singular cohomology this way, this will require some input that is not determined by π \pi_\infty.

Posted by: Peter Scholze on April 2, 2020 4:11 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thank you very much for the reply, and apologies for the inundation of questions! I’ll comment on a few things; I will (my apologies!) formulate a few further questions a little before the end which I’d very grateful if you would be able to take the time to think about, as if the construction which I will discuss works, it will be of some good for something I think.

Regarding π (X)\pi_{\infty}(X) versus the derived category of constructible sheaves: Using π (X)\pi_{\infty}(X) you can only see the locally constant constructible sheaves. If you want to see all constructible sheaves, you should use Barwick-Haine’s Gal(X)Gal(X), a condensed/pyknotic category, and then this is their exodromy theorem.

Actually, I was referring simply to the unbounded derived category of XX in the pro-étale topology, i.e. just complexes of sheaves of l\mathbb{Q}_{l}-modules, no constructibility assumption. I.e. I was claiming that this will be the homotopy category of the stable (,1)(\infty,1)-category given by Hom (,1)CAT)(π (X),H lMod)Hom_{(\infty,1)-CAT)}(\pi_{\infty}(X), H\mathbb{Q}_{l}-Mod).

In particular, I would emphasise that though one uses locally constant sheaves to construct π (X)\pi_{\infty}(X), one can certainly construct categories of other kinds of sheaves by mapping out of it. This is the same as for topological spaces: one can construct the anima (i.e. homotopy type) of a topological space XX using locally constant sheaves, but one can certainly construct various derived categories of XX by mapping out of the homotopy type of XX in a similar way to the above.

I am one of the few people who have never understood why one wants to consider the Nisnevich topology when one is studying motives. Note that rationally, everything is unchanged by working with the etale topology, and all interesting questions about motives (in the original sense) are rational (like Beilinson-Soule, etc.etc.).

It is a good point, I too in fact like to use the étale topology in a motivic setting. Nevertheless, I like the viewpoint on the Nisnevich topology as forcing that certain limits in schemes remain limits after localisation at the topology. This is a useful technical tool and conceptual crux if nothing else.

This would however know only about the Nisnevich cohomology of constant sheaves, and all interesting Nisnevich sheaves are not constant.

Here I would venture to remark that the point of using π (X)\pi_{\infty}(X) as I see it is to recover invariants of XX from purely homotopical gadgets, i.e. by mapping into condensed anima or similar. One cannot really criticise π (X)\pi_{\infty}(X) for not doing what it is built to do!

This brings me to the following, which I think is the heart of the matter here.

And I think you can’t possibly get algebraic singular cohomology this way, this will require some input that is not determined by π (X)\pi_{\infty}(X).

I would note that the Hom here is in the π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1})-local category, so there is definitely additional input here. If one works with model categories, one would have to take a fibrant replacement of K(A,n)K(A,n) in the π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1}) model structure if one wishes to compute the cohomology using a HomHom in condensed anima. I do not know whether we get exactly algebraic singular cohomology, but the picture is definitely close; the story is very similar to the usual one in motivic homotopy theory.

But this reminds me that what I really wanted to do for my ‘dream’ construction of algebraic singular cohomology/motivic cohomology was not exactly what I sketched before, but rather to replace the notion of étale homotopy of XX by some kind of 𝔸 1\mathbb{A}^{1}-local étale homotopy type, i.e. replace the condensed anima π (X)\pi_{\infty}(X) by some condensed anima π 𝔸 1(X)\pi_{\infty}^{\mathbb{A}^{1}}(X), so that algebraic singular cohomology really would be given by a Hom in condensed anima (without any localisation) from π 𝔸 1(X)\pi_{\infty}^{\mathbb{A}^{1}}(X) to K(A,n)K(A,n).

I am not aware of any attempt in the literature to construct something like π 𝔸 1(X)\pi_{\infty}^{\mathbb{A}^{1}}(X) even in the usual settings. But I am intrigued as to whether one can modify the Bhatt-Scholze construction to do something like this. In particular, I think that the construction of the fundamental group in section 7 of the paper might go through if one replaces sheaves on the pro-étale site in Definition 7.3.1 and everywhere else by 𝔸 1\mathbb{A}^{1}-local (in the usual sense) sheaves on the pro-étale site.

There are some schemes which are 𝔸 1\mathbb{A}^{1}-local, these are known as ‘rigid’ or ‘𝔸 1\mathbb{A}^{1}-rigid’.

In a bit more detail, as far as I see Lemma 7.3.9 immediately adapts to the 𝔸 1\mathbb{A}^{1}-local setting, and since 𝔸 1\mathbb{A}^{1}-local sheaves form a full subcategory which is complete and co-complete, and since sub-objects of 𝔸 1\mathbb{A}^{1}-local sheaves are obviously themselves 𝔸 1\mathbb{A}^{1}-local sheaves, I believe that it follows immediately from Lemma 7.4.1 and (in the case of (2) of Definition 7.2.1) its proof that we get an infinite Galois category in the 𝔸 1\mathbb{A}^{1}-local case too. Tameness is an immediate corollary of the tameness in the non-𝔸 1\mathbb{A}^{1}-local case.

If I am not overlooking something here, then presumably the same kind of adaptation to an 𝔸 1\mathbb{A}^{1}-local setting can be made in the (,1)(\infty,1)-case as well. And then we can consider what I really wish to consider: Hom(π 𝔸 1(X),K(A,n)Hom(\pi_{\infty}^{\mathbb{A}^{1}}(X), K(A,n) in condensed anima, without any 𝔸 1\mathbb{A}^{1}-localisation of the latter. Again, I suggest that this must be very close to some version of algebraic singular cohomology. If not, then what is it? Why should this story not follow the same pattern as the ordinary one for étale homotopy types?

I was wondering by the way if you could give some more details on the case of π \pi_{\infty} vs the construction of the fundamental group in the Bhatt-Scholze paper? For the latter does rely on some things that at first sight seem quite set-theoretic, e.g. the locally topologically Noetherian assumption. I suppose that one can replace all of the scheme theory/topology used by (,1)(\infty,1)-analogues, but it would be good if you can confirm whether this is what you have in mind, rather than some other construction?

I don’t think the Künneth formula will be true if one of the factors is 𝔸 1\mathbb{A}^{1} in either the pro-Nisnevich or the pro-etale settings; in the pro-etale setting, the Artin-Schreier problem in characteristic pp persists.

You are quite right, I got my wires crossed slightly here. One can formulate this as follows. What we should have is that π (X×𝔸 1)\pi_{\infty}(X \times \mathbb{A}^{1}) is equivalent to π (X)\pi_{\infty}(X). This is what should be called 𝔸 1\mathbb{A}^{1}-homotopy invariance. If it were true that π (X×𝔸 1)\pi_{\infty}(X \times \mathbb{A}^{1}) is equivalent to π (X)×π (𝔸 1)\pi_{\infty}(X) \times \pi_{\infty}(\mathbb{A}^{1}), then 𝔸 1\mathbb{A}^{1}-homotopy invariance would imply that 𝔸 1\mathbb{A}^{1} is contractible, which as you say is certainly not the case in characteristic pp.

Posted by: Richard Williamson on April 5, 2020 11:47 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

conceptual crux

conceptual crutch? It is was a conceptual crux you couldn’t just dismiss it! ;-)

Posted by: David Roberts on April 6, 2020 12:13 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

I believe condensed sets have been studied in constructive mathematics/type theory to interpret Brouwer’s fan theorem as sheaves over the Boolean Zariski site here.

A condensed set is a sheaf on the category of profinite sets with finite jointly surjective families of maps as covers.

Profinite sets are equivalent to Stone Spaces, and hence contravariantly equivalent to Boolean algebras. So a presheaf on ProFin is a functor from BA to Set.

The topology (finitely jointly surjective families) on profinite sets is equivalent to the Boolean Zariski topology.

For Pyknoticity, it seems possible to obtain a similar result by replacing Boolean algebras with normal distributive lattices, as they are a way of presenting compact hausdorff spaces with a finite cover topology, as emphasized in Johnstone’s Stone Spaces.

Provided I did not overlook something, this is an example where we can use pointfree topology to remove non-constructivity.

Posted by: Bas Spitters on March 31, 2020 10:50 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

A similar relation can be found in the work by Escardo and Xu. They observe that quasi-topopological spaces are the concrete sheaves of the big topos over endomorphisms of Cantor space.

Quasi-topological spaces are also “roughly” the concrete pyknotic sets.

Posted by: Bas Spitters on March 31, 2020 11:39 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

quasi-topopological spaces are the concrete sheaves of the big topos over endomorphisms of Cantor space.

No, not quite. Quasi-topological spaces are the small-set-valued concrete sheaves of the “very big” “topos” over all compact Hausdorff spaces (which is a large site — so in particular, a single quasi-topological space is a large object, even though concreteness implies that the category of quasi-topological spaces is locally small). They modify this definition by replacing “all compact Hausdorff spaces” by Cantor space.

Posted by: Mike Shulman on April 1, 2020 1:08 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks, I overlooked that yesterday late, but such size restrictions seem to be common when using topological sites. Escardo and Xu restricted to the separable case.

How crucial is it to work with large objects in the theory of pyknotic sets?

About cohesion, Mike already remarked that the topological topos is not cohesive. A similar argument seems to work here.

Posted by: Bas Spitters on April 1, 2020 6:34 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Mike, you made the connection to quasi-topological spaces in the nlab. I was about to ask whether this connection was known in the literature, but I see that the issue was already raised in the nforum.

Posted by: Bas Spitters on April 1, 2020 1:12 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

It is true that, if one wishes to avoid the axiom of choice, it may be better to replace profinite sets (=Stone spaces) with boolean algebras, and accordingly extremally disconnected profinite sets (=Stonean spaces) with complete boolean algebras.

However, I am not completely sure which theorems can be proved constructively. I’m using the axiom of choice ten times every page ;-).

In the arXiv reference you give, I think they use a different topology: The proetale topology is much finer than the big Zariski topology (which corresponds to open covers), and the difference is critical.

Also, the difference between pyknotic and condensed is not related to compact Hausdorff vs. profinite: As all compact Hausdorff spaces admit surjections from profinite sets, one can in both formalisms use either choice.

Posted by: Peter Scholze on April 1, 2020 9:58 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

What is the difference then? Is it only the choice of whether to bound the size of objects in the site at some “small” inacessible or to use a large site?

Posted by: Mike Shulman on April 1, 2020 10:27 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

It’s really just the set-theoretic issue: Either you cut off profinite sets (or compact Hausdorff spaces) at some (strongly inaccessible) cardinal κ\kappa, to get what Barwick-Haine call pyknotic sets and Clausen and I call κ\kappa-condensed sets, or you take the (large) filtered colimit over all κ\kappa of these categories, to get condensed sets.

What happened is the following: Clausen and I had called our things condensed sets, and then Barwick and Haine posted their paper, doing almost the same thing, but calling them pyknotic sets instead, and using a different way to resolve set-theoretic issues. We wondered how to proceed, and agreed that it’s best to use these two different names to refer to the slightly different ways to resolve the set-theoretic issues.

I agree that this is overemphasizing these set-theoretic issues, but as this causes some notable differences (such as the question of existence of injectives in abelian group objects) it may be alright; in any case better than having two teams of authors refer to exactly the same thing by different names without apparant reason.

Posted by: Peter Scholze on April 1, 2020 10:49 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Clausen and I had called our things condensed sets, and then Barwick and Haine posted their paper, doing almost the same thing, but calling them pyknotic sets instead, and using a different way to resolve set-theoretic issues. We wondered how to proceed, and agreed that it’s best to use these two different names to refer to the slightly different ways to resolve the set-theoretic issues.

Thanks for the history! We should add some notes along these lines to the nlab pages.

Is one of these approaches better than the other? Or do they both have advantages? Will one of them eventually become the standard? Or is there some way we could find a unified or mixed terminology to include them both in one theory? It would be nice if we didn’t have to maintain two similar theories that differ only in set-theoretic technicalities forever.

Posted by: Mike Shulman on April 4, 2020 6:18 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

This got longer than I intended, but maybe it’s helpful.

The advantage of the condensed approach is that it does not need universes, and in particular it is independent of any choice of (strongly inaccessible) cardinal κ\kappa.

Regarding language: Clausen and I define κ\kappa-condensed sets as sheaves on the site of κ\kappa-small profinite sets, for any strong limit cardinal κ\kappa. If κ\kappa is strongly inaccessible, these things are also called pyknotic sets (and the assumption of strong inaccessibility is often far too strong for what one is doing, strong limit cardinal is usually enough), and let me stress here that pyknotic sets depend on this implicit choice of κ\kappa. Now condensed sets are the union of κ\kappa-condensed sets along canonical fully faithful embeddings. These can also be defined as small sheaves on the category of all profinite sets – i.e. those sheaves that are small colimits of representable sheaves. So from the condensed point of view, one can easily talk about the pyknotic one by saying κ\kappa-condensed.

(Barwick and Haine have a different perspective, where they organize their choices of universes in a different way and end up seeing condensed sets as a subcategory of pyknotic sets. I admit that I find that perspective confusing – they first cut off their profinite sets at some strongly inaccessible κ\kappa, and then their condensed sets are the ones of V κV_\kappa, so they are certain sheaves on κ\kappa-small profinite sets with values in κ\kappa-small sets. So even what they call condensed sets depends on the choice of κ\kappa; arguably, they misuse the term.)

The advantage of the pyknotic approach is that you really have a topos, which comes with some benefits: You can apply adjoint functor theorems with no worries, your topos even has enough points, you have indiscrete pyknotic sets, …

My experience – and others may have different experiences – is the following: All the extra things you get in the pyknotic approach are not really helpful, and if you run into a situation where the difference between pyknotic and condensed becomes important, you’ve probably strayed off course. On the other hand, proving theorems is always slightly more difficult in the condensed setting; often, you first prove the result for κ\kappa-condensed sets and then verify that enlarging κ\kappa poses no problems. In other words, in the condensed setting one should only ever talk about objects that are compatible with “changing the universe” in the sense of enlarging κ\kappa, while the pyknotic approach is also allowing things that do depend on the choice of universe. It is in this sense that I regard constructions that work in the pyknotic case but not in the condensed case as pathological.

Let me discuss all of this in an example. One can define a functor XX̲X\mapsto \underline{X} from topological spaces to condensed sets, by setting X̲(S)=Cont(S,X)\underline{X}(S) = \mathrm{Cont}(S,X) for any profinite set SS. This defines a sheaf, as surjective maps of profinite sets (more generally, of compact Hausdorff spaces) are quotient maps.

Except that this is actually not a condensed set in general: It is a small sheaf (i.e., a small colimit of representable sheaves) if and only if all points of XX are closed, i.e. XX is T1T1. (This is not immediately obvious, but proved in the lecture notes.)

Restricting to κ\kappa-small profinite sets, X̲\underline{X} of course defines a κ\kappa-condensed set (in particular, a pyknotic set), and as I said above κ\kappa-condensed sets embed fully faithfully into condensed sets (via pullback), giving a condensed set X̲ κ\underline{X}_\kappa. However, the resulting condensed set X̲ κ\underline{X}_\kappa does not agree with X̲\underline{X} in general. If XX is T1T1, then for all sufficiently large κ\kappa, this is true. But if XX is not T1T1, the objects X̲ κ\underline{X}_\kappa do not stabilize when you increase κ\kappa.

So for any (strong limit) cardinal κ\kappa, one can build a functor XX̲ κX\mapsto \underline{X}_\kappa from topological spaces to condensed sets. Concretely, you take the above formula for κ\kappa-small profinite sets SS, and then take a left Kan extension to all profinite sets. On topological spaces with closed points, this functor is independent of the choice of κ\kappa if κ\kappa is large enough, but not otherwise.

My reading of this is that the functor XX̲X\mapsto \underline{X} should only be considered when XX has closed points. In the pyknotic approach, you are instead simply fixing your favourite (strongly inaccessible) cardinal κ\kappa, and use the functor XX̲ κX\mapsto \underline{X}_\kappa instead.

Now at first I was quite disappointed with this, as I’m a big fan of topological spaces with non-closed points (think of SpecA\mathrm{Spec}\, A for any ring AA; i.e., of spectral spaces). But there is a way to look at spectral spaces from the condensed point of view, which I think is part of Barwick-Haine’s exodromy equivalence, and Makkai’s conceptual completeness (see also the work of Lurie on this). Namely, spectral spaces embed fully faithfully into condensed categories (in fact, condensed posets), by taking their condensed category of points. (Similar results hold true more generally for coherent topoi, in which case you really need to talk about condensed categories, not just condensed posets.)

In other words, this information about specializations of points requires one to pass from condensed sets to condensed posets, explicitly making the specialization relations part of the structure.

Posted by: Peter Scholze on April 4, 2020 9:41 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Makkai’s conceptual completeness (see also the work of Lurie on this)

Since I was hoping my post would generate some discussion of conceptual completeness, I’ll mention some locations of Lurie’s work again: Ultracategories provides a new proof, and section A.9 of Spectral Algebraic Geometry develops the higher \infty-topos version.

Since this concerns reconstructing theories from models, what kind of theory is being reconstructed in the \infty case? [And does the logical scheme approach carry over?]

Posted by: David Corfield on April 5, 2020 8:13 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

So here’s something I’ve been wondering, and I should say it straight out: do condensed sets form an elementary topos? They form a cocomplete pretopos, but are there power objects? I vaguely conjectured yes, last year in discussions with Mike, but never got around to proving it either way. One way to approach it is via Theorem 2 in the notes from Topos à l’IHÉS available here. The pyknotic team of course say in print that condensed sets don’t form a topos, but are they assuming topos=Grothendieck topos? I note that in SGA4 Grothendieck et al discuss an example called ‘faux topos’, which is very much similar to the general setup you have.

The issue with injective abelian group objects in arbitrary elementary toposes is a known one. A quick search turned up a pair of papers by Roswitha Harting (Comm. Alg. 1982, JPAA 1983) looking into this. Blass (Trans. AMS 1979) gives an explicit example of a topos with no nontrivial injective abelian groups. So lack of enough injective abelian group objects is no obstruction to being a topos.

Posted by: David Roberts on April 1, 2020 1:38 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

For people working in algebraic geometry or neighboring fields, topos always means Grothendieck topos. To be honest, I don’t even know what an elementary topos is.

If power object means the internal Hom to {0,1}\{0,1\}, then yes, this exists – in fact, all internal Hom’s exist. Really the only difference to a Grothendieck topos is that the category is not generated by a set of objects.

PS: There are in fact no nonzero injective condensed abelian groups, see https://mathoverflow.net/questions/352448/are-there-enough-injectives-in-condensed-abelian-groups

Posted by: Peter Scholze on April 1, 2020 2:05 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

OK, I read the wikipedia entry on elementary topoi. The power object seems to be something different, classifying all subobjects (not just complemented ones). Somehow I’m having trouble picturing the power object of a point then.

So if I’m understanding it correctly now, the power object, even of a point, does not exist in condensed sets. The issue is that for any profinite SS, the subset S{s}SS\setminus \{s\}\subset S defines a subobject (for any sSs\in S), and this usually does not come via pullback from a smaller profinite set.

Posted by: Peter Scholze on April 1, 2020 2:34 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

So here’s a concrete argument: since condensed sets form a locally small category, if it had a subobject classifier Ω\Omega then Sub(X)Hom(X,Ω)Sub(X) \simeq Hom(X,\Omega) is a set. If you give me a condensed set that has a proper class of subobjects, then we have an obstruction to being an elementary topos. Would the terminal object work in this case? (it’s implicit in what you say)

There’s one more property that it would be good to know that CondCond has: being locally cartesian closed. If that is true, then we have that CondCond is a Π\Pi-pretopos, which is almost the next best thing to being a(n elementary) topos. Then it would be good to check if all WW-types exist, I imagine using (Swan 2018), but I’m a bit suspicious that the axiom WISC might not hold, which that paper requires.

Posted by: David Roberts on April 1, 2020 10:14 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Any given condensed set only has a set worth of subobjects. The issue is more stuble – the subobject classifier could be defined as a sheaf on all profinite sets, but it would not be a small sheaf (i.e. is not generated under small colimits by the sheaves represented by profinite sets). If it was small, then any subobject of a profinite set SS would come via pullback frome such a subobject of a profinite set of bounded cardinality, and this is not true.

On the positive side, condensed sets are locally cartesian closed, and they do satisfy WISC if I understand that axiom correctly. The point is that any condensed set admits a surjection from a disjoint union of Stonean spaces, and any surjection onto a disjoint union of Stonean spaces splits.

What are W-types?

Posted by: Peter Scholze on April 1, 2020 10:35 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

any surjection onto a disjoint union of Stonean spaces splits.

Any epimorphism from a condensed set?

What are W-types?

WW-types are generalisations of natural number objects. They are useful for talking about free algebraic gadgets of an inductive nature. For instance, they allow us to talk of the free monoid object on an object of generators, or the free internal category on an internal graph object. Talking about arbitrary WW-types is big generalisation, though. They also allow for transfinite recursion.

Posted by: David Roberts on April 1, 2020 11:20 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Yes, I meant “epimorphism from a condensed set”.

That being said, I realize I probably misunderstood WISC; I don’t really know what “in the internal logic” means.

Regarding W-types: Condensed sets are the large filtered colimit of the topoi of κ\kappa-condensed sets along the left-adjoint pullback maps of topoi. This means that whenever one does a “free” construction inside condensed sets, then all intervening condensed sets are κ\kappa-condensed sets for some κ\kappa, and one can do the construction in κ\kappa-condensed sets (which form a topos). As the functors are left adjoints, they should commute with “free” constructions, and hence this should still be free in κ\kappa'-condensed sets for all κκ\kappa'\geq \kappa, and thus in condensed sets.

From what you write, W-types sound like free constructions, so could an argument along these lines work?

Posted by: Peter Scholze on April 2, 2020 7:34 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

It seems like the subobject classifier exists, but lives in a larger universe. This is also what happens for hSets in predicative homotopy type and it is analogous to what happens with the object classifier in higher topos theory. The object classifier is a small universe, and hence lives in a larger universe. Likewise, the classifier for nn-truncated maps lives in a higher universe. By “accident” (impredicatively) the subobject classifier, the classifier for -1-truncated maps, lives in the small universe.

A similar point was made by Mike in 2011.

Does this seem correct for condensed sets?

Posted by: Bas Spitters on April 2, 2020 2:02 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Yes, that’s right.

Posted by: Peter Scholze on April 2, 2020 3:54 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

What can we conclude about pyknotic or condensed \infty-groupoids (anima) as to whether they form an (,1)(\infty, 1)-topos, elementary or Grothendieck?

Posted by: David Corfield on April 10, 2020 2:43 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Pyknotic anima are, by definition, a Grothendieck (,1)(\infty,1)-topos.

Condensed anima are not a Grothendieck (,1)(\infty,1)-topos. They do not admit a subobject classifier (for size reasons). They do seem to satisfy the predicative version of the definition of an elementary (,1)(\infty,1)-topos given in that nLab entry, except possibly for the existence W-types – I would have to learn what those are before I can comment.

Posted by: Peter Scholze on April 10, 2020 8:50 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks! So, at least in the case of the former, those HoTT-derived results hold, such as the Blakers-Massey theorem, and indeed for any ‘modality’ the generalized Blakers-Massey Theorem.

Posted by: David Corfield on April 11, 2020 8:22 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Marc Hoyois and Urs Schreiber have a conversation about cohesion and pro-étale sites from here. They cover the lack of product-preservation of the extra left adjoint, as well as other issues discussed above.

Posted by: David Corfield on April 2, 2020 7:18 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Ah, indeed! Thanks for finding this.

As maybe a small further remark: I think the failure to be product-preserving should also be there in characteristic 00, although this is slightly more subtle. I’ll add a comment if I find some concise argument.

Posted by: Peter Scholze on April 2, 2020 7:51 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Just a quick couple of remarks. Firstly, unless there are some unexpected differences between the étale and pro-étale worlds, ‘after ll-completion’ it should be true that π ()\pi_{\infty}(-) preserves products. I.e. probably one could work not with condensed anima but with ll-completed condensed anima everywhere if one needed this.

As a second remark, one should get product preservation in some cases I think even without ll-completion, e.g. when one of the schemes involved is 𝔸 1\mathbb{A}^{1} (𝔸 1\mathbb{A}^{1}-homotopy invariance). This is related to some of my earlier comments, e.g. this one. May I ask whether you saw/got a chance to think about this one by the way, Professor Scholze?

One further remark that occurs to me, by the way, about π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1})-homotopy as defined in my earlier comment, is that, in contrast to the naïve version of 𝔸 1\mathbb{A}^{1}-homotopy, the naïve version of π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1})-homotopy equivalence is I believe already an equivalence relation, because one can use the \infty-groupoid structure to ‘reverse’ and ‘compose’ π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1})-homotopies.

Morevoer, the cohomology theory Hom(π (X),K(A,n))Hom(\pi_{\infty}(X), K(A,n)) for some XX and some AA in this π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1})-local (,1)(\infty,1)-category, which I suggest must be similar to algebraic singular cohomology, is definitely π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1})-local (this follows from some adjunction juggling). In other words, using π (𝔸 1)\pi_{\infty}(\mathbb{A}^{1}) instead of 𝔸 1\mathbb{A}^{1} seems to erase the distinction between naïve and non-naïve 𝔸 1\mathbb{A}^{1}-homotopy theory. This is an important technical point I believe.

Posted by: Richard Williamson on April 2, 2020 3:25 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

One of the motivations for studying the discrete functor and its two adjoints is that the fracture squares of differential cohomology follow for stable objects.

I wonder if when you say above:

Condensed anima have two directions, a homotopy-theoretic direction (anima) and an actual topological space direction (condensed),

is this a form of such fracturing for the monad and comonad, f *f !f_{\ast} f_{!} and f *f *f_{\ast} f^{\ast}?

Posted by: David Corfield on April 2, 2020 8:58 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

From Peter’s Lectures on Analytic Geometry, p. 76

there are two natural functors from CW complexes to condensed anima: A fully faithful functor, via the full subcategory of condensed sets; and another non-faithful functor, factoring over the full-∞-subcategory of anima. Let us (abusively) denote the first functor by XXX \mapsto X, and the second by X|X|X \mapsto |X|. Then S 1S^1 is a physical circle, while |S 1||S^1| is some ghostly appearance of a point with an internal automorphism (the anima B).B \mathbb{Z}).

That kind of phenomenon is very much modal HoTT territory, with the distinction between the topological 0-type, the circle 𝕊 1\mathbb{S}^1, and the (discrete) higher inductive type, S 1S^1 (using the notation of Mike’s paper). The ‘shape’ of the former is equal to the latter.

In a sense, I suppose, this is arising from the cohesive relation between topological ∞-groupoids and ∞-groupoids, before inserting topological ∞-groupoids into condensed ones.

Posted by: David Corfield on April 11, 2020 12:30 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

David, thanks for all these pointers and questions!

Yes, this is very closely related. Note that on the same page of the analytic lectures (Lemma 11.9) there are given some natural conditions on condensed anima for which the left adjoint exists – e.g., CW complexes, in that case given by the ‘shape’ – as Dustin also discussed in some comment above. Unfortunately, the class of all condensed anima for which this left adjoint exists has no good closure properties – it is of course stable under all colimits, and it contains the interval [0,1][0,1] (thus all CW complexes) but it is not closed under finite limits (as Cantor sets can be written as an equalizer of two maps [0,1][0,1][0,1]\to [0,1]).

Having a framework that can handle topological and homotopical structures at the same time is of course extremely convenient and somewhat surprisingly within the algebraic topology literature there does not really seem to be a good language for doing so. Mike Shulman’s paper you cited seems to do so; are there other papers in this direction? (I guess cohesive \infty-topoi are some kind of axiomatization of this situation.)

Posted by: Peter Scholze on April 11, 2020 8:55 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

I think the most substantial use of this idea of combining the homotopic and the topological (or geometric) using the shape and discrete (flat) modalities is in Urs’s setting of the differential cohomology diagram.

It applies in any case of relative cohesion, so broadly, even if the use of terms such as ‘differential forms’ suggests otherwise. Some exploration of this breadth is given in a talk, differential cohesion and idelic structure, he gave.

There’s also been some work on a synthetic theory of infinitesimal cohesion, as in Felix Wellen’s article.

Posted by: David Corfield on April 12, 2020 8:25 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

There are two preprints containing basic algebraic topology results using Mike Shulman’s framework (real-cohesion). They both start with abstract modal type theory and have some applications in the end mixing topological and homotopical structure:

Modal Descent by Egbert Rijke and myself. It has some basic covering-space theory as an application. This turns out to be quite concise in real-cohesion.

Good Fibrations through the Modal Prism by David Jaz Myers. This one goes further and has more applications in a similar direction.

Even with basic algebraic topology topics, when looking at it in real-cohesion, there are always generalizations and questions in plain eye sight that would never come to my mind in any classic presentation of the topic. In the latter above, there is a generalization of covering spaces (from level 1) to arbitrary homotopy levels. Starting at level 2, the (2-)universal covering space stops being trivial in the homotopy-direction, and this turns out to be something that just does not work in plain topology and collapses to n-connected covers in plain homotopy theory.

Posted by: Felix Cherubini on April 17, 2020 12:09 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Starting at level 2, the (2-)universal covering space stops being trivial in the homotopy-direction, and this turns out to be something that just does not work in plain topology and collapses to n-connected covers in plain homotopy theory.

For what it’s worth, one might consider checking out my PhD thesis, Fundamental bigroupoids and 2-covering spaces. The tech is pretty low-brow, I would do it slightly differently these days

Posted by: David Roberts on April 18, 2020 12:47 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

It’s been a long time coming out as a paper, but the Borsuk-Ulam theorem has also been proven in real-cohesion.

The really nice thing about cohesion is that so much of the geometric structure can be captured by the pure categorical structure of an adjoint string (or multiple such strings, in the case of enhancements like differential cohesion), and moreover that that adjoint string can be fairly easily represented in the internal type theory via modalities. The lack of suitable adjoints in the pyknotic/condensed situations makes it sound like they would be much less convenient in that way. Are there different categorical structures that can be used to capture the relationship between topology and homotopy theory in the pyknotic/condensed world?

Posted by: Mike Shulman on April 17, 2020 10:19 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Hi Mike,

Let me apologize in advance for writing yet another long post. It was short in my head, I promise! By the way, please don’t take all of this verbiage and these explanations, which certainly veer off from the topic you brought up, as me proselytizing; I have respect for all the different formalisms under discussion and have no desire to persuade or dissuade anyone from using whichever or whatever they’d like in their work. I’m simply fascinated by the structres which have emerged in our study and excited to share!

I think the best one can do in the pyknotic/condensed setting is what I described above: a partially-defined left adjoint to the pullback from the terminal topos. This tells you which objects “have a homotopy type” or are “locally contractible”, and outputs correct homotopy type on them. I don’t think any kind of categorical structures in our setting can change this basic fact that by design there just simply are objects which don’t have a homotopy type (but only a pro-homotopy type).

While it seems less convenient to only have a partially defined left adjoint, in my experience one can work with it just fine. Also, I would like to give 3 reasons (though there are surely more as well) why it’s important for us to allow these non-locally contractible spaces into the basic set-up.

First: one of our main goals was to develop a clean formalism for analytic geometry which works both in the non-archimedean and archimedean cases. The kinds of topological rings that show up in non-archimedean analysis are just not locally constractible.

This point can be made quite concretely. Suppose you have a discrete ring R. You can view it either as a discrete condensed ring or as discrete real-cohesive ring. Now take an ideal I in the ring, and form the completion (inverse limit of R/I^n) internal to whichever setting you’re working in. In the condensed setting, you will get the condensed ring associated to the usual topological completion. This is no longer discrete; in fact it naturally recovers the topological completion. But in the real cohesive setting, essentialy by virtue of the left adjoint always being defined, any inverse limit of discrete objects is discrete, so you at most see the I-adic completion with its discrete topology. This is insufficient for our purposes, and here you really see that the desire to be cohesive and the desired to incorporate non-archimedean analysis are straightforwardly incompatible.

Second: this point is much more deep and shocking, and speaking for myself I can’t say that I fully understand it. It turned out for us that even if we only pretend to be interested in locally contractible spaces, more precisely real manifolds, we still need to allow non-locally contractible building blocks, essentially these profinite sets we base everything on.

This looks like a ridiculous claim in view of the fact that everybody and their mother has successfully studied real manifolds and noone has used profinite sets to do it (or at least not systematically), but we wanted to have a natural abelian category whose objects have the same nature as “topological real vector spaces”, but whose categorical properties are almost just as nice as usual modules over a discrete ring. (This is the story of “liquid modules” described in Peter’s notes, and which we touched on above.) We found that to produce this category and prove its basic properties it was necessary to deform the real numbers to a ring of (overconvergent) Laurent series with integer coefficients, whose structure is that of a countable union of profinite sets rather than being locally contractible.

This is very much related to thinking of real numbers in terms of say binary expansions; note that the space of binary expansions of elements of [0,1] is profinite, being a countable product of copies of {0,1}. It seems that the fact that real numbers are fundamentally built from profinite sets was “there all along” right under our eyes, we had just chosen to ignore it.

Third: the last reason is structural. Profinite sets are in many ways just simpler than real intervals. To say basically the same thing in many ways: they have a basis of clopen subsets, they are 0 dimensional, the theory of sheaves on them is purely combinatorial (reduces to finite sets), etc. This, plus the existence of enough “projective” profinite sets (the extremally disconnected spaces), translates into a very nice property of the category of condensed sets: it is generated by compact projective objects. This means that besides behaving like a topos it also behaves like an algebraic theory. This is absolutely fundamental to our desire to have our categories of analytic modules behave in similar ways, categorically speaking, to usual modules over usual rings.

Posted by: Dustin Clausen on April 18, 2020 11:52 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

This means that besides behaving like a topos it also behaves like an algebraic theory.

Reminiscent of Joyal and Anel’s logoi:

The notion of topos can be similarly presented as dual to the algebraic notion of logos. (Topo-logie)

Posted by: David Corfield on April 18, 2020 8:55 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

I do have to say I’m curious what you mean by “behave similarly to usual modules over usual rings”. Internal ring objects in any topos, of course, behave very much like usual modules over usual rings – including cohesive toposes. The main difference is that you have to use constructive logic. So is the point of the profinite world that you can retain more classical logic?

Posted by: Mike Shulman on April 18, 2020 10:07 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks for the examples, Dustin! I definitely appreciate that there is value to non-locally-contractible spaces. I’m just wondering whether we can find a different categorical structure (not, of course, a totally defined adjoint) that captures the relationship between geometry and homotopy in that world.

Put differently, what does the internal language of the topos of pyknotic/condensed \infty-groupoids look like? Can we internalize the notion of “pro-homotopy-type” somehow? Is it sufficient to be able to talk about maps from topological objects into homotopical ones without necessarily having a representing object for them?

Can you give any simple examples of proofs in this world that we could try to write purely internally and see what structure they require?

Posted by: Mike Shulman on April 18, 2020 10:05 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Dear Mike,

this is in reply to your last two comments, both of which are about the internal logic of condensed sets. I must admit that while I do get the hang of “working internally in a topos” and quite enjoy doing so, I’ve never been able to get all the way. In particular, I don’t understand what the “internal logic” is, and what it means to describe this “internally”. Can you give me some relevant pointers? Better yet, some concrete examples?

Maybe the following comment has some relevance. In condensed mathematics, one does indeed often feel the drive to work internally, and thus to “condense” everything, and hence even pass to condensed categories – so that one is not talking anymore about categories of modules over a ring, but about condensed categories of modules. This whole formalism works to some degree, but there are also things that break; notably, the most desirable way to make solid abelian groups into a condensed category runs into some issues (but it works for solid p\mathbb{Z}_p-modules for any prime pp, curiously). Part of the reason is that the proof that solid abelian groups behave well uses a theorem of Nöbeling saying that the continuous integer-valued functions on any profinite set form a free abelian group; the proof makes very heavy use of the axiom of choice, and is the kind of proof that’s impossible to do internally in any other topos (and, as I implied, its relevant generalization fails). Generally, I’ve decided that it’s best (for me, at least) to try to resist the drive to work completely internally. But I might also be wrong about this.

Regarding the other question, “why do modules over analytic rings behave similarly to usual modules over usual rings”: There are two aspects, one related to the topos of condensed sets, and one related to the notion of an analytic ring. The first aspect is that the topos of condensed sets has compact projective generators, which implies the same for the category of modules over any condensed ring. This is definitely not true in any topos – in fact, in quite few of them. It is not too far from some kind of axiom of choice in your topos – while not any surjection splits, any condensed set XX admits a surjection from some X˜\tilde{X} such that any surjection onto X˜\tilde{X} splits. I think this last statement is really a key statement you gain in comparison to cohesive settings.

The other aspect is about the notion of an analytic ring. This is meant to encode the idea of “complete” modules over a condensed ring; in particular, the tensor product is automatically a “completed” tensor product. The remarkable thing here is that all the nice categorical structures (a closed symmetric monoidal category of modules, existence of compact projective generators, etc.) pass to this subcategory of “complete” modules. This, I believe, is something really novel, as usually completion of modules does not have such very nice categorical properties.

Posted by: Peter Scholze on April 19, 2020 11:31 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Well, I don’t have time to write out lengthy examples here, but I can try to give some pointers. The “internal logic” is a formal system that’s adequate to be a foundation for mathematics, but also admits a “nonstandard” interpretation where the “types” become objects of the topos. By describing something “internally” I mean to write it in this formal system in such a way that when the formal system is interpreted in the topos the result is the original topos-theoretic description. In this paper I tried to sketch the idea of internal logic for a topologically-minded reader. Some relevant examples of internal reasoning are the results in cohesive HoTT that were mentioned above.

the most desirable way to make solid abelian groups into a condensed category

This is an interesting phrase. From the perspective of the internal logic of the topos of condensed sets, there is no choice of a “way” to make anything into a condensed category. If the definition of that category can be written in the formal internal language system, then the interpretation of that definition automatically yields an internal category in – or, in the case of a large category, an indexed category over – the topos. So, for instance, abelian group objects in condensed sets automatically form an indexed category over condensed sets (the analogue for large categories of an internal category in condensed sets). If “solidness” can be expressed internally, the same is true for solid abelian groups. It would be interesting to know how this relates to your “most desirable way”.

any condensed set XX admits a surjection from some X˜\tilde{X} such that any surjection onto X˜\tilde{X} splits.

This is indeed a form of the axiom of choice, known in constructive circles as the presentation axiom or “COSHEP”. It plays an important role in certain kinds of constructive mathematics, so it’s very interesting to see it arising here as well. In particular, it means that a larger amount of “constructive mathematics” can be internalized in condensed sets than in an arbitrary topos (e.g. Bishop’s constructive mathematics, and that of his followers, can often be formalized in a context where the presentation axiom holds).

Can you say anything comprehensible about what aspect of condensed sets it is that causes the nice behavior of completion?

Posted by: Mike Shulman on April 20, 2020 11:21 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Your paper on internal logic is a nice read, thanks for the pointer!

That link on the presentation axiom in particular says that CoSHEP implies WISC, answering an earlier question of David Roberts: Condensed sets satisfy WISC. I’m pretty sure they admit W-types, but didn’t come around to checking that carefully.

About the question on how to turn solid abelian groups into a condensed category: You see, I’m old. There is a platonic ZFC universe of sets around me, and all my statements and constructions happen there.

To discuss the question, let me recall the definition of solid abelian groups. First, for any profinite set S=lim iS iS=\lim_i S_i (with S iS_i finite sets), let [S] =lim i[S i]\mathbb{Z}[S]^{\blacksquare} = \lim_i \mathbb{Z}[S_i], as a condensed abelian group. This comes with a map [S][S] \mathbb{Z}[S]\to \mathbb{Z}[S]^{\blacksquare}, where [S]\mathbb{Z}[S] is the free condensed abelian group on the condensed set SS.

Now a condensed abelian group MM is solid if any map [S]M\mathbb{Z}[S]\to M extends uniquely to [S] M\mathbb{Z}[S]^{\blacksquare}\to M.

Theorem. The class of solid abelian groups is stable under all limits and colimits, and extensions, … . It is also the minimal subcategory of condensed abelian groups that contains \mathbb{Z} and is stable under all limits and colimits. Its compact projective generators are I\prod_I \mathbb{Z}, for any set II.

How would this internalize? I think this means that I have to do the same construction in any slice topos. It’s probably enough to do it in a slice over a profinite set TT. Then one can do a relative version of the construction of [S] \mathbb{Z}[S]^{\blacksquare} – write any profinite set SS over TT as a limit of “relative discrete + finite” S iS_i over TT, and take the corresponding limit in the slice topos. Asking the above condition would then produce what I consider the most desirable way to make solid abelian groups into a condensed category. Unfortunately, it does not really work. The main reason is that the limit is not equal to the derived limit, and one should be taking the derived limit. (The non-existence of higher inverse limits in the absolute case relies on that theorem of Nöbeling I mentioned earlier. When working with p\mathbb{Z}_p-coefficients, one even has the vanishing relatively, essentially because one is taking a limit of compact Hausdorff objects.) I didn’t check this carefully, but it may be true that when passing to derived categories, things are again OK; but one will definitely lose some properties.

Here is another, very important, cautionary word about working internally vs. externally in condensed sets. As discussed above, condensed sets have enough compact projective objects. These are exactly the extremally disconnected profinite sets. However, these objects are not internally compact projective! In other words, the internal Hom does not commute with all sifted colimits. The issue at hand is that a product of two extremelly disconnected profinite sets is no longer extremally disconnected. This causes quite a headache at times. (Coming back to the start of this message, condensed sets do not satisfy internal CoSHEP.)

You had one other question: What causes the nice behaviour of completion? That’s a great question, but I’m not sure I’m able to say anything sensible. Definitely the exactness properties (of infinite products etc.) play a very important role, as do various compactness arguments – like that one has a coherent topos, but also again the existence of compact projectives, etc.

Posted by: Peter Scholze on April 21, 2020 10:30 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks for the explanation!

condensed sets do not satisfy internal CoSHEP

That’s too bad. However, I expect it should still be possible to state a version of “\flat-CoSHEP” that would hold in their internal language augmented by a modal operator, analogous to the “\flat-LEM” and “\flat-AC” that hold in some cohesive models.

How would this internalize? I think this means that I have to do the same construction in any slice topos.

More or less, but there’s some room for discussion in the meaning of “the same”. A fully internal construction would be to have some internally-indexed family of maps, say a map f:ABf:A\to B of abelian groups in a slice category over some object II, and consider in every slice the category of abelian groups that are internally-local with respect to “all fibers of” ff. For maps of mere types, people have studied this kind of internal localization in HoTT (e.g. RSS and CORS).

If we took II to be the discrete object on the set of profinite sets and ff the family of your maps [S][S] \mathbb{Z}[S] \to \mathbb{Z}[S]^{\blacksquare}, this would almost agree with your definition in the global case. The only difference would be that the localization would be internal, meaning that the induced map of condensed sets Hom([S] ,M)Hom([S],M)Hom(\mathbb{Z}[S]^{\blacksquare},M) \to Hom(\mathbb{Z}[S],M) would have to be an isomorphism. Sometimes that extra condition is automatic; is it in this case?

It’s not clear to me that this choice of II is the best one, though, when considering other slices. Is there a “classifying condensed-type of profinite sets”?

Posted by: Mike Shulman on April 23, 2020 4:55 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Does the “\flat-CoSHEP” just concretely mean that any surjection onto a discrete set splits? That’s definitely true, even internally.

It is true that for a solid abelian group MM, the map Hom([S] ,M)Hom([S],M)\mathrm{Hom}(\mathbb{Z}[S]^{\blacksquare},M)\to \mathrm{Hom}(\mathbb{Z}[S],M) of internal Hom’s is an isomorphism. This is not obvious, but a general property we ask for “analytic rings”. It implies that solidification defines a symmetric monoidal functor (for a unique symmetric monoidal tensor product on solid abelian groups).

You could indeed in the relative case only ask for being local with respect to the pullbacks of the maps [S][S] \mathbb{Z}[S]\to \mathbb{Z}[S]^{\blacksquare}, but as you say this seems a bit awkward. The condition I proposed in the previous comment amounts to asking the same for what you call “the classifying condensed-type of profinite sets”, which I take to mean the stack associating to each profinite set SS all profinite sets T/ST/S.

I think that’s the right condition, but as I said it doesn’t work on the abelian level (but plausibly does on the derived level).

By the way, there are reasons to care about this besides foundational things. This precise setup, associating to any profinite set the “derived category of solid sheaves of abelian groups over it” (or maybe with \mathbb{Z}_{\ell}-coefficients), can be extended to any condensed anima by descent, and then defines some kind of 66-functor formalism. (Interestingly, the functor Rf !Rf_! (and Rf !Rf^!) fails to exist, but as a substitute for Rf !Rf_! one gets a left adjoint of f *f^\ast in complete generality (satisfying base change and projection formula) – this ties the discussion back to the early questions about cohesion –, and this left adjoint can often play the role of Rf !Rf_!.) A key property one gains by imposing solidity is Poincare duality: For a compact submersion f:YXf: Y\to X, the left adjoint to f *f^\ast is just a shift and twist of the right adjoint Rf *Rf_\ast. This is related to Dustin’s comment that the solidification of [X]\mathbb{Z}[X] for a CW complex XX is given by [|X|]\mathbb{Z}[|X|] where |X||X| is the corresponding anima (and if XX is a smooth manifold, this is the same as RΓ(X,)R\Gamma(X,\mathbb{Z}) up to shift and twist).

These things can also be done for schemes, or for diamonds, and are relevant in my upcoming paper with Laurent Fargues on the geometrization of the local Langlands correspondence.

Posted by: Peter Scholze on April 23, 2020 9:36 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Something to raise if anyone wants to take the cohesion idea further here, perhaps taking \infty-sheaves on profinite sets as the base, is that it may be possible to find a further layer of structure in the form of a second layer of adjoints. This constitutes what is called differential cohesion.

With this structure in place, a good deal of mathematics follows: deformation theory, de Rham spaces, jet bundles, partial differential equations. Urs develops the later in particular with Igor Khavkine in Synthetic geometry of differential equations.

I wonder how much follows with slightly less than full (differential) cohesion.

At the nForum we bumped up against such issues with Buium’s arithmetic jet spaces in his arithmetic differential geometry and then with Borger’s absolute geometry.

Not that I understand very much in it, but it’s interesting to see deformation theory appear in Purity for Flat Cohomology.

Posted by: David Corfield on April 5, 2020 10:22 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Regarding the possible “other” adjoint to the pullback from sets to condensed sets, meaning the left adjoint, there is an interesting story here concerning the interplay between topology and homotopy theory, or “condensation” and “animation” if you will.

To see it we should switch from sets to anima and from condensed sets to condensed anima. As discussed above, the left adjoint to the inclusion of anima into condensed anima doesn’t actually exist. But it’s better to say instead that it’s defined on some objects and not others. Thus, for a condensed anima XX, you can ask whether or not there exists an initial anima |X|\vert X\vert with a map X|X|X \to \vert X\vert.

The answer is sometimes yes, and sometimes no. For the absolute most basic condensed anima, namely the profinite sets, the answer is no unless the profinite set is finite. Indeed it’s easy to see that there is always an inintial pro-anima with a map from any X, and for profinite sets this pro-anima is just the corresponding pro-finite set.

Since the answer is no for these most basic objects, you might think it’s pretty much always no. But interestingly enough, this is not the case: these profinite anima can collude together via even very simple colimits to produce honest anima.

Here is an example. We can view any CW complex X as a condensed set, hence as a condensed anima. For such X there is an initial anima with a map from X, and it is the usual anima associated to the CW-complex.

Thus this formalism can tell you which topological spaces (or condensed sets, or condensed anima) really deserve to have a homotopy type (or anima) attached to them. The profinite sets do not, but CW complexes do. It also makes clear and precise the philosophical point that “a topological space maps to its anima”, which the usual perspective of the associated singular simplicial set does not.

(The perspective provided by the shape theory of higher topoi does, however, in much the same way. The theory of higher topoi and the theory of condensed anima have many such similarities. A big difference is that the \infty-category of condensed anima is much better behaved than the \infty-category of \infty-topoi: it has compact projective generators, and it is itself an \infty-topos modulo cardinality issues.)

It’s interesting to try to unwind this claim explicitly. Any finite CW complex, being a compact Hausdorff space, has a simplicial resolution by profinite sets. The claim is that if you take the geometric realization of this simplicial object in the \infty-category of pro-anima, you get a constant anima with value the anima we all know and love associated to that finite CW-complex (e.g. as modelled by its singular simplicial set).

Posted by: Dustin Clausen on April 5, 2020 11:28 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

This is such a lovely point of view, Dustin.

Is there a reasonable functor from condensed anima to infty-topoi that connects the two not-quite-left-adjoints (namely your |-| and the shape)? (Of course profinite anima embed fully faithfully into infty-topoi, so I suppose one can just try to extend this embedding. I wonder how destructive that would be.)

Posted by: Clark Barwick on April 5, 2020 11:51 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Hello, Clark! Nice to run into you, as always, even if only online. Though I guess that’s how people run into each other these days anyway.

Yes, there is such a reasonable functor from condensed anima to \infty-topoi. It sends a condensed anima XX to the \infty-topos X etX_{et} of “etale sheaves” over X (c.f. Scholze’s diamonds paper for the analogous construction there). There are two nice descriptions of this. One is the one you suggest: take the usual \infty-category of sheaves of anima on the (extr. disc.) profinite sets, and Kan extend from there to general XX. Thus, an etale sheaf on a general XX is a compatible family of etale sheaves on all profinites mapping to XX. Another description is that it identifies with a certain full subcategory of CondAn /X\operatorname{CondAn}_{/X} closed under finite limits and colimits (and pullbacks), namely the one spanned by the “etale” maps to XX. I’ll skip the precise definition of “etale”.

In the examples I’ve had occasion to think about, this association XX etX\mapsto X_{et} is quite reasonable and not very destructive. For a (T_1) topological space TT, there is a pullback map from the usual \infty-topos of sheaves of anima on TT to the \infty-topos of etale sheaves on the associated condensed anima. For locally compact Hausdorff space, or a CW-complex, this pullback map identifies the target with the Postnikov completion of source. Similarly, for BG, the classifying anima of a profinite group, the \infty-topos of etale sheaves identifies with the Postnikov completion of the “usual” \infty-topos of sheaves on finite continuous GG-sets with the canonical Grothendieck topology.

On the other hand, I expect that in general XX etX \mapsto X_{et} is not fully faithful. One test would be to see whether X etX_{et} knows the difference between the filtered colimit of all compact at most countable subspaces of [0,1][0,1] and [0,1][0,1] itself. Actually, this is a question that probably many people here at the cafe could weigh in on:

Question: Is the pullback functor from sheaves of anima on the interval [0,1][0,1] to compatible families of sheaves of anima on all compact at most countable subspaces of [0,1][0,1] an equivalence?

The answer to the analogous question on the level of 00-topoi (i.e. locales) or on the level of topological spaces is negative, basically because the topology on [0,1][0,1] is sequential. But it’s not clear to me what’s going on for \infty-topoi or even 11-topoi. I found it kind of tricky to analyze, which is one of several reasons why a few years ago I abandoned \infty-topoi for this kind of thing and switched to the condensed perspective. Of course I’ve also learned many times over the years that just because I find something tricky or difficult doesn’t necessarily mean that it really is so…

Finally, regarding your question about shape theory, note that on the “discrete” condensed anima XX, meaning the usual anima, X etX_{et} is the usual slice topos of anima over XX. Thus, returning to general XX, for formal reasons we get a comparison map of pro-anima Shape(X et)|X|\operatorname{Shape}(X_{et})\rightarrow \vert X\vert. Again this is an equivalence for CW complexes, compact Hausdorff spaces, etc., but probably not in general.

One last remark. Since XX etX\mapsto X_{et}, viewed as a functor to \infty-topoi with geometric morphisms as morphisms, is left Kan extended from the very simple profinite sets, it feels like it “wants” to be a left adjoint. But there is no right adjoint, for reasons like the one Peter (Scholze) mentioned: if your \infty-topos is at all “category-like”, e.g. admits non-trivial specializations between points, then the associated condensed anima won’t exist and the associated pyknotic space will likely be pathological.

It seems that the difference in nature between condensed anima and \infty-topoi is twofold:

  1. \infty-topoi can be “category-like” whereas condensed anima can only be “space-like”;

  2. \infty-topoi are based on abstracting “second-order” structure: open subsets of topological spaces. Condensed anima are based on abstracting “first-order” structure: points of topological spaces.

I guess it is fundamentally these two differences which give rise to all the other differences. For example the relative simplicity of the \infty-category of condensed anima vs. the \infty-category of \infty-topoi is heavily influenced both by 1 and 2. And the fact that condensed anima are good for mixing algebraic and topological structures (whereas \infty-topoi are not) is clearly due to point 2.

I also believe (though do not have honest evidence to support this belief, so take it with a grain of salt) that these differences mean that comparisons between natural constructions associated to condensed anima and \infty-topoi, such as their associated pro-anima as discussed above, can only work out in certain nice special cases and not in general. Thankfully, however, the nice special cases do include many objects of general interest.

Posted by: Dustin Clausen on April 6, 2020 10:15 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

I’m confused: When you write “negative answer” above, do you mean “positive answer”?

In any case, I believe the map *B\ast \to B\mathbb{R} of condensed anima induces an equivalence on associated X etX_{\mathrm{et}}’s – a contractible group like \mathbb{R} must act trivially on any discrete anima.

Whether your other example also works, I don’t really know. Also, it feels wrong to me to think of condensed anima as certain \infty-topoi – to me, \infty-topoi have always been quite different beasts. Probably as my examples of interest are things like the etale site of a scheme, which both has specializations of points, and automorphisms of points, none of which exists for condensed anima. On the other hand, as stated before, there is this relation between certain (\infty-)topoi and certain condensed (\infty-)categories, which is also in some way generalizing the construction you are talking about. Do you see how to fit everything together?

Posted by: Peter Scholze on April 6, 2020 10:50 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Sorry, of course condensed anima can have automorphisms of points! Delete that part of the message…

Posted by: Peter Scholze on April 6, 2020 10:52 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Hi Peter!

Yes, of course, you’re right about B*B\mathbb{R}\rightarrow \ast inducing an equivalence on () et(-)_{et}. That fact just didn’t occur to me when I was writing my comment. Thanks for the reminder.

And I agree it doesn’t make sense to try to view condensed anima as certain \infty-topoi. That’s what I was trying to say at the end of my comment.

I also agree that it would make more sense to compare condensed \infty-categories and \infty-topoi: at least, this gets rid of the difference 1 from the end of my previous comment. But because of difference 2 I still don’t think there’s a truly general comparison available. (Though it works in the suitably “coherent” setting I guess by Jacob’s work.)

Posted by: Dustin Clausen on April 6, 2020 11:15 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Hi Peter!

Yes, of course, you’re right about B*B\mathbb{R}\rightarrow \ast inducing an equivalence on () et(-)_{et}. That fact just didn’t occur to me when I was writing my comment. Thanks for the reminder.

And I agree it doesn’t make sense to try to view condensed anima as certain \infty-topoi. That’s what I was trying to say at the end of my comment.

I also agree that it would make more sense to compare condensed \infty-categories and \infty-topoi: at least, this gets rid of the difference 1 from the end of my previous comment. But because of difference 2 I still don’t think there’s a truly general comparison available. (Though it goes through in the suitably “coherent” setting I guess by Lurie’s ultracategory work.)

Posted by: Dustin Clausen on April 6, 2020 11:17 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Sorry for the double-post above. Also, yes, I meant “positive answer” when I wrote “negative answer”. (It means the question of fully faithfulness has a negative answer, which is why I mixed the two up…)

Posted by: Dustin Clausen on April 6, 2020 11:20 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

I wonder if this is related to Johnstone’s topological topos, which is a category of sheaves on a very disconnected site (roughly, \mathbb{N}_\infty). Two of its important properties are that the internally built CW-complexes have the correct topology, and that its singular complex / geometric realization adjunction to simplicial sets is a geometric morphism of toposes.

Posted by: Mike Shulman on April 23, 2020 4:57 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

I’m pretty sure that these properties also hold for condensed sets. In some sense condensed sets are the version of Johnstone’s topos that replaces convergence along sequences by convergence along ultrafilters; the difference should not be relevant here. (Let me instead point out that Johnstone’s topos does not have enough projective objects, and in a way condensed sets are the minimal (up to cardinality questions) way to make them have enough projective objects.)

Posted by: Peter Scholze on April 23, 2020 9:41 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks for the pointer, Mike! That fact about geometric realization is nice and I never new it. Confirming Peter’s assertion, I did check that it works in condensed sets as well. The point is just that pushouts of compact Hausdorff spaces where one maps is a closed inclusion, as well as filtered unions of topological spaces where every compact subset lives in some finite level, are preserved by the functor from topological spaces to condensed sets. And the geometric realization of every simplicial set can be “resolved” by these in a simplicial way. Combined with Joyal’s observation about the topos of simplicial sets classifying interval objects and the fact that [0,1] is an interval object in condensed sets, this does it.

By the way, besides the non-existence of compact projectives in Johnstone’s topological topos, one can also mention the fact that it is not even coherent. Coherence is of course a very basic finiteness property for topoi. In particular, in setting up the solid theory we use all over the place that compact Hausdorff spaces are coherent condensed sets.

Posted by: Dustin Clausen on April 23, 2020 10:26 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Before I suggested the connection with topological sites and continuous truth. A key difference is that there one only allows open covers.

The internal logic of the topological site over compact Hausdorff spaces, or compact regular locales, is well-understood. For example, we have that all functions from Cantor space (2 N2^N) to the Booleans are uniformly continuous. I believe this should continue to hold for Pyknotic sets.

In topological sites, we also have a principle of local choice: if we have a total predicate on a suitable internal space, we obtain a cover and choice functions on that cover. This one fails for Johnstone’s topological topos, as we can cover N N_\infty by the even E E_\infty and odd O O_\infty numbers (including \infty).

Ch III.2,3 in Johnstone’s Stone spaces show that every compact Hausdorff space can be covered by a Stone space (which is projective), and explains the relation with the double negation modality that Mike mentioned. If my understanding is correct, this internal Stone locale will be “representable” and hence have enough points, as in the theory of topological sites.

As Mike mentioned, in Johnstone’s topological topos, the unit interval [0,1] is an interval. The key property to check is that [0,1] is covered by [0,x] and [x,1] for every x. This is also true in Pyknotic sets, so should also have a geometric morphism from simplicial sets to Pyknotic sets which coincides with the geometric realization.

Posted by: Bas Spitters on April 23, 2020 10:55 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

David Ben Zvi just sent me a link to this discussion. I’m sorry to have been silent; we just weren’t aware of it. At this point, I don’t have much to add, except to say the following:

(1) I think Dustin and Peter S understood a lot about this stuff well before we did. (We were just oblivious to it till Jacob Lurie told us to contact Dustin.)

(2) Our aims are largely – but not completely – different. Our goal was to extend the exodromy theorem to the context of Qell or Qell-bar coefficients and beyond. That works: constructible Qell-bar sheaves (say) ‘are’ pyknotic functors from Gal(X) to Perf(Qell-bar). This is proved in the latest version of exodromy on our websites in section 13 – http://math.mit.edu/~phaine/files/Exodromy.pdf. (We’ll arXiv soon; just clearing out typos.)

(3) The theorem I’ve alluded to in (2) should follow from the result that Cisinski kindly cited, but Peter H and I have had a hard time filling in the details of proof for that without making a story that was far too long. (Peter H is finishing his PhD next year, so a little more efficiency is maybe called for.)

(4) I agree (I won’t speak for Peter H) with everything Peter S has said about the pyknotic/condensed distinction. In particular: if you’re working with a pyknotic object that isn’t condensed, you may be contemplating a pathology. (In fact, in the earliest version of the pyknotic paper, we worked only with accessible sheaves, but we stopped because we wanted to be indolent in our use of the adjoint functor theorem.)

Posted by: Clark Barwick on April 5, 2020 10:53 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

We have a case of relative cohesion discussed at the nLab. The entry for differential algebraic K-theory understands

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

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

Posted by: David Corfield on April 8, 2020 8:20 AM | Permalink | Reply to this

Condensed vs Bornological

One other topic we didn’t mention is Lawvere’s bornological topos, finite product-preserving presheaves on the category of countable sets.

Although the bornological topos can be regarded as a cohesive category of “spaces” in a broad sense, it doesn’t satisfy Lawvere’s axiomatic cohesion since it lacks the required left adjoint components functor.

Compare

“functional analysis becomes linear algebra” in the bornological topos

and

The bornological topos is the Gaeta topos on CC and as such fits Lawvere’s paradigm of doing abstract “algebraic geometry”

with

The goal of this course is to launch a new attack, turning functional analysis into a branch of commutative algebra, and various types of analytic geometry (like manifolds) into algebraic geometry. (Lectures in analytic geometry)

Posted by: David Corfield on April 11, 2020 2:32 PM | Permalink | Reply to this

Re: Condensed vs Bornological

Let me try to answer.

First, I think Lawvere’s choice to stick to countable sets was just to avoid the kind of set-theoretic nonsense that has bugged much of this thread. I think the comparison to the condensed/pyknotic setup is cleaner if one simply takes the bornological site to consist of all sets, and covers finite families of jointly surjective maps (and resolves set-theoretic issues in the same way as before).

There is a functor from condensed sets to bornological sets, taking a condensed set XX to the functor that takes a set AA to X(βA)X(\beta A) where βA\beta A is the Stone-Cech compactification of AA. This is (modulo set-theoretic issues) the pushforward functor for a map from the condensed topos to the bornological topos. The difference between the two setups is that in condensed sets, X(βA)X(\beta A) is functorial in continuous maps βAβA\beta A\to \beta A' (equivalently, in maps AβAA\to \beta A' of sets), and there are more such maps than those induced by maps of sets AAA\to A'. If you wish, condensed sets are bornological sets equipped with some extra structure related to “limit points”, where limits are understood in terms of ultrafilters: If XX is the bornological set coming from a condensed set X 0X_0, and you have an element xX(A)x\in X(A) for some set AA (i.e., intutively, “a bounded AA-sequence x=(x a) aAx=(x_a)_{a\in A} of elements of XX”) then for any ultrafilter 𝒰\mathcal{U} on AA, one can take the limit of x ax_a along a𝒰a\in \mathcal{U} to get a new element lim 𝒰xX(*)\mathrm{lim}_{\mathcal{U}} x\in X(\ast). Namely, 𝒰\mathcal{U} defines a point of βA\beta A, and you have a map X(A)=X 0(βA)X 0(*)=X(*)X(A) = X_0(\beta A)\to X_0(\ast) = X(\ast). So in this sense condensed sets know quite a bit more about analysis, as they also know about (some kind of) limits.

Posted by: Peter Scholze on April 11, 2020 9:17 PM | Permalink | Reply to this

Re: Condensed vs Bornological

Something’s off here regarding the interpretation of the above functor as a pushforward in a map of topoi. The left adjoint would send AA to βA\beta A, but this does not commute with finite products as β(A×B)βA×βB\beta(A\times B)\neq \beta A\times \beta B.

On the other hand, one can regard the above functor as a pullback in a map of topoi, so giving a map from the bornological topos to the condensed topos. (I’m ignoring set theory here, so either pass to the pyknotic setup, or replace “topos” by “macrotopos”.) This is induced by the map of sites taking any profinite set SS to the bare set SS.

This functor evidently loses a lot of information, but it is conservative: If f:XYf: X\to Y is a map of condensed sets that is an isomorphism on associated bornological sets, then in particular X(βA)Y(βA)X(\beta A)\cong Y(\beta A) for all sets AA, and the Stone-Cech compactifications of discrete sets form a basis for the pro-etale site of a point.

Posted by: Peter Scholze on April 12, 2020 12:19 AM | Permalink | Reply to this

Re: Condensed vs Bornological

The drive to turn some or other branch of mathematics into “algebraic geometry” is interesting. Breiner, in the work I’ve referred to a couple of times already, writes

Although contemporary model theory has been called “algebraic geometry minus fields”, the formal methods of the two fields are radically different. This dissertation aims to shrink that gap by presenting a theory of logical schemes, geometric entities which relate to first-order logical theories in much the same way that algebraic schemes relate to commutative rings.

Whereas Lurie’s approach to this model-theoretic topic in Ultracategories uses ultraproducts to manage convergence, Breiner has his models form a topological groupoid.

I wonder if there’s a place for condensed mathematics in model theory.

Posted by: David Corfield on April 12, 2020 12:05 PM | Permalink | Reply to this

Re: Condensed vs Bornological

This is probably orthogonal to the interests of everyone else on this thread, and yet another instance of a functional analyst being silly by taking seriously comments like “turning functional analysis into a branch of commutative algebra”, but are the “Smith spaces” in the notes you link to the same as what Cigler, Losert and Michor called Waelbroeck spaces? (There’s an adjunction between Ban and Ban^{op} implemented by the duality functors and I have a vague recollection that Wael identifies with Ban^{op}.)

More generally: I wonder how much of the drive I’ve seen in recent years to embed Ban in a “better” category overlaps with the old ideas that people like Noel and Waelbroeck were espousing back in the 1970s and 1980s? Waelbroeck was basically, AFAICT, trying to embed Ban into the heart of its derived category, which is an abelian category, and work with the result in a hands-on way.

(The Ben-Bassat–Kremnitzer approach, or the small aspect of it that nonchalantly parks its tanks on the lawn I “grew up with”, seems to be doing something similar; it rather makes me wish someone had informed me and my PhD supervisor back in 2004 that the thing we’d read in the literature and knew applied to Ban would in fact be something taken seriously by Proper Mathematicians.)

Posted by: Yemon Choi on April 12, 2020 7:04 AM | Permalink | Reply to this

Re: Condensed vs Bornological

Hi Yemon,

Thanks for your comments and questions. I find the interaction with functional analysis fascinating. Let me start with some quick responses to your questions/musings; if you’d like me to say more about any of these points I’d be very happy to expand. I believe they’re also all discussed in Peter’s notes on analytic geometry.

1) Smith spaces are the same as Waelbrock dual spaces. Smith’s work came much earlier; it proved the main theorem on them (the dualilty with Banach spaces), but did not specifically isolate the concept. We got the name from Akbarov.

2) Smith spaces are more natural than Banach spaces from the condensed perspective because they are controlled by a nice compact subset instead of a nice open subset (the unit ball) like in the Banach setting. Also, Banach spaces are filtered colimits of Smith spaces while Smith spaces are filtered limits of Banach spaces; filtered colimits are nicer in their algebraic and homological properties so it makes sense to take Smith spaces as fundamental.

3) But they are not fundamental enough! Smith spaces embed fully faithfully in condensed R-vector spaces; this is good, but Smith spaces are not an abelian category, and we also care about things like Ext’s. There is an induced functor from Waelbrock’s enlarged (abelian!) cateogry of quotients of Smith spaces, but that functor is not fully faithful!

4) The lack of full faithfuness in 2) is not accidental; it arises because Waelbrock’s rather formally defined category does not “see” certain natural functional analytic maps and constructions, specifically those like Ribe’s famous extension which are based on the entropy functional. These extensions take place outside the familiar comforting context of locally convex spaces, so it is tempting to exclude them “by fiat” as basically everyone does implicitly or explicitly, but from the condensed perspective they are simply “there” and must be reckoned with. Besides, their structre is fascinating and leads to a one-parameter family of deformations of the real numbers, as described in Lecture VI of Peter’s notes.

5) The conclusion is that if you want a naturally ocurring abelian category suitable for functional analysis you absolutely need to allow non-locally convex spaces in addition to the familiar locally convex ones.

6) Fortunately such an abelian category exists; in fact there is a one-parameter family of such categories. These are the “p-liquid R-modules”. They sit naturally in condensed R-modules, have excellent categorical and homological properties, include essentially all examples of functional analytic spaces people use, and calculations such as tensor products, hom’s, ext’s etc. work out nicely with them.

6) More specifically, p-liquid R-modules are an example of what Peter and I define as an “analytic ring”. This is an axiomatic set-up consisting of a condensed ring plus a full subcategory of condensed modules over that condensed ring, which one can to first approximation think of as the “completete” modules. The axioms are very strong and force the category of complete modules to behave much in the same way as ordinary modules over an ordinary ring. All the honest work lies in producing examples of such analytic rings.

7) Proving that the p-liquid theory “exists” (satisfies the axioms of an analytic ring) is very difficult and requires a delicate mixture of homological algebra and functional analysis, with implicit constants several layer deep floating around everywhere. See lectures VIII and IX of Peter’s notes. But once you have it you can take it as a black box and many results in complex analytic geometry prove themselves “by hand” with categorical techniques, plus simple calcuations in key cases.

Best regards Dustin

Posted by: Dustin Clausen on April 12, 2020 9:59 AM | Permalink | Reply to this

Re: Condensed vs Bornological

Dear Yemon,

thanks for your question, and for allowing yourself to be silly! In fact, I’m sorry for this totally over-the-top first paragraph or two of my lectures, but I wanted to make the ambition clear. And I actually think that the theory developed in the first 9 lectures is something of real value even to actual functional analysts, and is actually going some way towards the ambition.

As Dustin has explained in his long comment, proving our results about liquid R-vector spaces requires very serious efforts (it might be the most difficult theorem I’ve (co-)proved), both conceptually but also just the bare estimates. But once you have it you can easily do a lot of things in complex analysis and potentially even in other situations like Atiyah-Singer where you usually run into difficult technical issues of doing homological algebra with topological vector spaces. The machinery we developed is supposed to gracefully handle such problems. We’ve mostly worked out some things for complex-analytic spaces so far, but there’s nothing stopping this approach from working for smooth manifolds and the like.

Posted by: Peter Scholze on April 12, 2020 10:15 PM | Permalink | Reply to this

Re: Condensed vs Bornological

I was just looking through some of Waelbrock’s work, and indeed it seems he’s also done other things which go in a similar direction as our work. I see that he and Buchwalter defined a notion of “compactological space” which turns out to be equivalent to the notion of “quasi-separated condensed set”. (Quasi-separatedness is a general topos-theoretic concept; here it unwinds to the following: a condensed set X is quasi-separated iff the diagonal map X –> X x X is a closed inclusion, in the sense that every base-change to a profinite set identifies with a closed inclusion of profinite sets. It is the natural separation axiom to impose in the condensed context.) Indeed, both are equivalent to the full subcategory of Ind(compact hausdorf spaces) spanned by the Ind-objects which can be represented using only closed inclusions for transition maps.

Posted by: Dustin Clausen on April 14, 2020 12:33 PM | Permalink | Reply to this

Codensity monads

I’ve been expecting someone to bring up the topic of codensity monads, one of Tom Leinster’s favourite topics.

Above, we’ve had some left adjoints not appearing, which is often the occasion for a codensity monad to appear. As Tom puts it,

Even a functor without a left adjoint induces a monad, just as long as certain limits exist. This is called the codensity monad of the functor.

The ultrafilter monad is a codensity monad, something Clark Barwick mentions in his talk, Who cares about pyknosis?.

The ultraproduct construction is also a codensity monad, Section 8 of Tom’s Codensity and the ultrafilter monad. It seems that some enrichment is needed to describe ultracategories (p. 20 of Richard Garner’s Ultrafilters, finite coproducts and locally connected classifying toposes).

Posted by: David Corfield on April 13, 2020 10:49 AM | Permalink | Reply to this

Codensity monads

My 2-category theory is rather primitive, but does this sound right?

Implicitly, there’s a (weak) 2-monad on CatCat operating in Ultracategories, one which sends a category MM to the category of ultrastructures on MM. An algebra for this 2-monad is an ultracategory. An ultrafunctor (left ultrafunctor, right ultrafunctor) between ultracategories is an algebra morphism (colax, lax).

And this 2-monad is the codensity 2-monad for the inclusion of finite sets into CatCat as discrete categories.

Clearly I’m just trying to imitate the construction of the ultrafilter monad.

Posted by: David Corfield on April 14, 2020 6:16 PM | Permalink | Reply to this

Re: Codensity monads

I think I can prove at least one of the statements above. Give me time to write down what I have in mind.

Posted by: Fosco Loregian on April 15, 2020 6:36 PM | Permalink | Reply to this

Re: Codensity monads

I think I can prove at least one of the statements above. Give me time to write down what I have in mind.

Posted by: Fosco Loregian on April 15, 2020 6:37 PM | Permalink | Reply to this

Re: Codensity monads

If I understand well what you’re after, you want to show that a ultracategory is just a TT-algebra for a suitable monad.

My claim is that the monad acts as follows on objects:

T:𝒜 Xβ(X)×𝒜 X T : \mathcal{A} \mapsto \int^X \beta(X)\times \mathcal{A}^X

I am basing this conjecture on the fact that an ultracategory consists of maps

i X:β(X)×𝒜 X𝒜 i_X : \beta(X)\times \mathcal{A}^X \to \mathcal{A}

that if this conjecture is correct amount exactly to a functor T𝒜𝒜T\mathcal{A} \to \mathcal{A} (by the universal property of the coend, the i Xi_X forming a cowedge in XX).

(Sure, we could debate about where exactly I’m taking the coend in question… β(X)\beta(X) is a topological space, and 𝒜 X= xX𝒜\mathcal{A}^X = \prod_{x\in X} \mathcal{A} is a category; I see different, inequivalent ways to give a meaning to the product β(X)×𝒜 X\beta(X) \times \mathcal{A}^X

A deeper problem is: strictly speaking TT does not exist, because as XX ranges over all sets the coend I’m taking is on too large a diagram. I kindly ask you to set aside size issues and follow a bit of coend-nonsense: so, just for today TT exists and defines a functor.

  1. First, the initial cowedge i X:β(X)×𝒜 XTAi_X : \beta(X) \times \mathcal{A}^X \to T A has a component at X=1X=1, the terminal category; this gives TT the structure of a pointed functor (of course, ii depends naturally on 𝒜\mathcal{A}, thus we have a natural transformation η:1T\eta : 1 \Rightarrow T).

  2. A candidate multiplication map TT𝒜T𝒜T T\mathcal{A} \to T\mathcal{A} can be obtained as follows:

TT𝒜 = Yβ(Y)×T𝒜 Y = Yβ(Y)×( Xβ(X)×𝒜 X) Y Y Xβ(Y)×β(X) Y×𝒜 X×Y Y Xβ(Y)×ββ(X) βY×𝒜 X×Y ϵ Y Xββ(X)×𝒜 X×Y μ β Y Xβ(X)×𝒜 X×Y colim Y Xβ(X)×𝒜 X×Y Xβ(X)×𝒜 X\begin{array}{rl} T T\mathcal{A} &= \displaystyle \int^Y \beta(Y) \times T\mathcal{A}^Y \\ &\displaystyle = \int^Y \beta(Y) \times \left( \int^X \beta(X)\times \mathcal{A}^X \right)^Y \\ &\displaystyle \to \int^Y\int^X \beta(Y) \times \beta(X)^Y \times \mathcal{A}^{X\times Y} \\ &\displaystyle \to \int^Y\int^X \beta(Y) \times \beta\beta(X)^{\beta Y} \times \mathcal{A}^{X\times Y} \\ &\displaystyle \overset{\epsilon}\to \int^Y\int^X \beta\beta(X) \times \mathcal{A}^{X\times Y} \\ &\displaystyle \overset{\mu^\beta}\to \int^Y\int^X \beta (X) \times \mathcal{A}^{X\times Y} \\ &\displaystyle \to \text{colim}_Y\int^X \beta (X) \times \mathcal{A}^{X\times Y} \\ &\displaystyle \cong \int^X \beta (X) \times \mathcal{A}^{X} \end{array}

All this is pretty standard (ϵ\epsilon is the evaluation, counit of the cartesian closed structure on… wherever β\beta takes values; μ β\mu^\beta is the multiplication of the monad β\beta); the last step is “colimits over a category with a terminal object are evaluation at that object”.

Of course, there’s still some work to do now: the map μ:TT𝒜T𝒜\mu : T T\mathcal{A} \to T\mathcal{A} is associative and has η\eta as a unit. I just wanted to provide quantitative support to your initial guess.

Posted by: Fosco Loregian on April 15, 2020 7:18 PM | Permalink | Reply to this

Re: Codensity monads

Thanks, Fosco. Yes, I just wanted a description of ultracategories as algebras for a 2-monad. Since in the case of discrete categories, ultracategories are ultrasets, and these are compact Hausdorff spaces (Sec 3.1), this seems to be a method to impose some kind of compact topology on a category.

Posted by: David Corfield on April 16, 2020 10:23 AM | Permalink | Reply to this

Re: Codensity monads

David, I hadn’t commented before because I’ve only given “Ultracategories” the briefest skim, but given our conversation over here, I’ll do a bit of thinking out loud.

In short, I don’t get the idea. First let’s do what you suggest but with 1- rather than 2-monads. The codensity monad of your inclusion FinSetCatFinSet \hookrightarrow Cat sends a category AA to the discrete category on the set of ultrafilters on π 0(A)\pi_0(A). Here π 0(A)\pi_0(A) denotes the set of connected-components of AA. (This follows from the adjunction π 0D\pi_0 \dashv D.) And presumably that’s not the right thing.

So if your proposal is right, something significant must happen when we step up from 1-monads to 2-monads. Can you explain what that is?

Posted by: Tom Leinster on April 15, 2020 7:34 PM | Permalink | Reply to this

Re: Codensity monads

Thanks, Tom. Perhaps, the first thing is to ascertain whether or not we’re dealing with a 2-monad.

If so, it would be pleasant to give some basic construction of it.

Posted by: David Corfield on April 15, 2020 8:55 PM | Permalink | Reply to this

Codensity monads

More idle musings:

Ultracategories are pseudopyknotic (is there ‘pseudocondensed’?) categories (Sec 4.3 of Pyknotic objects, I), where the ‘pseudo’ has to do with the difference between sheaves and hypersheaves. So which ultracategories are pyknotic?

2-monads would bring us to the topic of doctrines, categorified theories.

Jacob Lurie develops the concept of an ‘ultracategory envelope’, for use in the \infty-version of conceptual completeness. I wonder if higher-monad theory could make sense of this.

It would be interesting to explore higher codensity monads for all kinds of inclusion of (discrete) compact things, finite (finitely presented) sets/categories/\infty-groupoids, etc. into various codomains.

Posted by: David Corfield on April 15, 2020 12:18 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Let me try to give a slightly different answer to Mike’s questions.

First of all, I’m not used to thinking about the internal language of a topos. But still I believe it is fair to say, as you suggest, that in the condensed world you retain more classical logic as opposed to in cohesive settings. I tend to think of this more in terms of what properties of sets (or abelian groups, or anima, or…) pass to condensed sets (etc). The existence of compact projective generators means that any isomorphism claim involving sifted colimits and (arbitrary) limits reduces to the analogous claim in sets (or anima). For abelian groups you can even drop the “sifted”. Combined with the usual topos theoretic fact that finite limits interact with colimits in the same way as in sets (etc), this means that condensed X’s inherit essentially all exactness products from ordinary X’s. E.g. infinite products and infinite direct sums of condensed abelian groups are exact. Probably, however, this discussion covers only one aspect of “working internally”.

About internalizing “pro-homotopy type”, I’m not entirely sure what it means, but I’ll just note that it’s not true that pro-anima embed fully faithfully into condensed anima. Various weakenings of this statement do hold, but the general pro-anima are just too wild. On the other hand I do think it’s suficient to just talk about maps to discrete anima without carrying around a representing object.

You asked for a relevant proof to try to internalize in this context; again, I’m not comfortable enough with internalization to necessarily offer a good example, but maybe this will be interesting. It is a proof that the unit interval is contractible.

Claim: The initial discrete anima with a map from the condensed set (associated to) [0,1] is *\ast.

Proof: Think of [0,1] as glued from two copies of itself placed end-to-end. In condensed sets (and even condensed anima) this gives a pushout diagram with one of the maps a monomorphism. Now iterate, gluing [0,1] from 2 n2^n copies of itself, and pass to the inverse limit over nn. Inverse limits commute with such pushout diagrams in condensed anima, so we get a pushout diagram at the infinite level. But in the limit all the terms we’re gluing from become profinite. Thus for purposes of mapping to discrete anima we can pass back down to the finite level, but replacing each [0,1] by just *\ast. Then you easily see the contractibility even at finite level. QED

As for our categories of modules over analytic rings behaving like modules over usual rings, as Peter says the point is the existence of enough compact projectives. Again, this means the same exactness properties as abelian groups. Note also that an abelian category with a single compact projector is the same as the category of modules over a ring. I do want to point out one key difference, which concerns the tensor structure (say, in the commutative case): the compact projectives are not always closed under tensor product. Fundamentally this comes from the fact, a continual source of technical troubles in this theory, that a product of two extr. disc. profinite sets is almost never itself extr. disc. The more obvious difference that the tensor unit (though compact projective) does not generate the category should also be kept in mind.

Let me bring up another point related to this. Abelian categories with enough compact projectives can be manufactured very cheaply and in abundance: they are just the same as categories of additive functors from some arbitrary additive category to abelian groups. For example, our category of solid abelian groups, which is our base category for non-archimedean analysis, can be simply described as the category of additive functors from free abelian groups to arbitrary abelian groups. Note that this is not how we define it: we define it as a full subcategory of condensed abelian groups, and it takes a lot of work to prove enough about this category to see that it is equivalent to the above category of functors.

Now, why not just use this simple definition and avoid all the work necessary to embed it in the condensed context? There are actually several reasons, one being the rather important, though non-mathematical, fact that we would never have conceived of this solid theory or its possibilities without the condensed context guiding us there. But another is that the interaction with condensed structures is interesting and useful in and of itself. Let me give an example of this which will in fact come full circle and give a partial answer to your original question about replacing this left adjoint with a different construction (whether or not this construction classifies as “categorical” is highly debatable, however).

Namely, I claim that if you’re willing to pass to suspension spectra, then you do get a fully well-defined functor which takes a CW-complex (viewed as a condensed set) to the suspension spectrum of its associated anima. But now this is not a functor to ordinary spectra, but a functor to condensed spectra which happens to take “discrete” values (discrete in the condensed direction) on CW-complexes.

Actually, what you really have is a colimit-preserving endofunctor of condensed spectra, which when applied to the suspension spectrum of a CW complex (viewed as a condensed set) gives the suspension spectrum of its associated anima. This is the “solidification” functor, which works for spectra much in the same way it works for abelian groups. On the suspension spectra of profinite sets varinjlimX i\varinjlim X_i, it takes the value varinjlim𝕊[X i]\varinjlim \mathbb{S}[X_i]. This is not discrete, but for quite non-obvious reasons it once more nonetheless gives discrete values on CW complexes.

This solidification functor is also what gives the “correct” completion functor in the non-archimedean context. So it is a completely globally-defined functor which implements non-archimedean completion, and in the archimedean setting implements passage to homotopy type. The existence of such a thing is quite novel as far as I’m aware and powerful in my opinion, especially because it has extremely good formal properties.

Posted by: Dustin Clausen on April 19, 2020 12:53 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

That description of the [0,1][0,1] interval put me in mind of Freyd’s coalgebraic characterisation.

Posted by: David Corfield on April 19, 2020 3:04 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Yes, thank you, David! That’s a really nice characterization. I guess I was implicitly using it, or a variant of it which says that [0,1] is the only compact Hausdorff space II equipped with two endpoints and a gluing isomorphism from two copies of itself glued along the endpoints to itself, such that a point in II is uniquely determined by a sequence of letters “L” and “R” saying in which half of the decomposition the point lies, which half of that half, etc. etc. Not nearly as crisp as Freyd’s characterization, but it’s what the proof uses :)

Posted by: Dustin Clausen on April 19, 2020 3:58 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Dustin, there’s a topological version of Freyd’s characterization that might fit what you’re doing. It goes like this.

Let’s say a bipointed space is a topological space equipped with an ordered pair of distinct, closed points. Given bipointed spaces (X,x ,x +)(X, x^-, x^+) and (Y,y ,y +)(Y, y^-, y^+), one can form their wedge XYX \vee Y in the usual way (take their disjoint union, identify x +x^+ with y y^-, and take x x^- and y +y^+ to be the new basepoints). Let CC be the category in which an object is a bipointed space X=(X,x ,x +)X = (X, x^-, x^+) equipped with a continuous, basepoint-preserving map XXXX \to X \vee X. For example, ([0,1],0,1)([0, 1], 0, 1) together with the doubling map is an object of CC.

Theorem The terminal object of CC is [0,1][0, 1], with its standard (Euclidean) topology.

(Of course, if you stick to Hausdorff spaces then you can drop the condition that the basepoints are closed.)

The nice thing about this theorem is that you don’t get some trivial topology on [0,1][0, 1]. You get the Euclidean topology. And the result can be understood as saying that [0,1][0, 1] is uniquely suitable as a parametrizing space for homotopy theory, as explained here.

This theorem isn’t mentioned in the nLab page that David linked to. It follows from Theorem 2.2 of my paper arXiv:1010.4474. There, the result is embedded in a much wider theory of self-similarity, but it’s also not hard to prove directly.

Posted by: Tom Leinster on April 19, 2020 4:38 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

P.S: I mistyped in the above proof: filtered inverse limits commute with pushouts where one maps is an isomorphism only for coherent condensed anima, not for general condensed anima.

Posted by: Dustin Clausen on April 19, 2020 4:16 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

…. and in the above post: “monomorphism” instead of “isomorphism”. Sorry all! My brain’s a bit fried these days.

Posted by: Dustin Clausen on April 19, 2020 4:17 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Thanks for this! I’d like to understand this proof about the unit interval better. Can you write it out in more detail? In particular, I don’t understand what you mean by “pass to the inverse limit” and “at the infinite level”, or why profiniteness means that for mapping into discrete types we can “pass back down to the finite level”.

You call the solidification of spectra a “colimit-preserving endofunctor”. Is it idempotent? Is it a reflector or coreflector?

Posted by: Mike Shulman on April 23, 2020 5:08 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

The solidification of spectra is completely parallel to the story of solid abelian groups discussed above. It is an idempotent endofunctor, and is the left adjoint to the inclusion of solid spectra into condensed spectra. I’m always getting confused about signs, so I don’t know whether this makes it a reflector or coreflector.

About passing to finite level: If S=lim iS iS=\lim_i S_i is a profinite set, written as a (co)filtered limit of finite sets, then for any discrete anima XX, considered as a condensed anima via the fully faithful embedding, one has X(S)=colim iX(S i)X(S)=\colim_i X(S_i). This property in fact characterizes discrete objects: A condensed anima XX is discrete if and only if for all profinite SS as above, X(S)=colim iX(S i)X(S)=\colim_i X(S_i).

I’m also a little confused about the precise inverse limit Dustin wants to take, but here is a variant that definitely works.

For any nn, let I nI_n be the disjoint union of the intervals [i/2 n,(i+1)/2 n][i/2^n,(i+1)/2^n] for all i=0,,2 n1i=0,\ldots,2^n-1. Then f n:I nI=[0,1]f_n: I_n\to I=[0,1] is surjective, and one can recover II as the geometric realization of the Cech nerve I n /II_n^{\bullet/I} of f nf_n.

For varying nn, there are natural transition maps, and so let I =lim nI nI_\infty = \lim_n I_n. Sequential limits of surjections are still surjections in condensed sets – this uses the existence of compact projectives. So f :I If_\infty: I_\infty\to I is still a surjection, and one can recover II as the geometric realization of the Cech nerve I /II_\infty^{\bullet/I}.

But now observe that I I_\infty (and more generally all terms of the Cech nerve) is profinite; in fact, the natural map

I lim nI nlim nπ 0(I n)I_\infty\to \lim_n I_n\to \lim_n \pi_0(I_n)

is an isomorphism. This is because all the intervals are shrinking in size. In fact, I I_\infty gets naturally identified with binary expansions of elements [0,1][0,1], without identifications such as 0.011111...=0.10000...0.011111... = 0.10000..., so I = {0,1}I_\infty = \prod_{\mathbb{N}} \{0,1\}.

Similarly,

I /Ilim nI n /Ilim nπ 0(I n /I)I_\infty^{\bullet/I}\to \lim_n I_n^{\bullet/I}\to \lim_n \pi_0(I_n^{\bullet/I})

is an isomorphism.

Now let XX be any discrete anima; for simplicity truncated (in order to commute some limit with a sequential colimit in a second). Then one can compute Hom(I,X)\Hom(I,X) as the limit of the cosimplicial object Hom(I /I,X)\Hom(I_\infty^{\bullet/I},X). As each term is profinite, each term is the sequential colimit of

Hom(π 0(I n /I),X)\Hom(\pi_0(I_n^{\bullet/I}),X).

This sequential colimit can be exchanged with the limit, so it is enough to compute for each nn the limit of the cosimplicial object

Hom(π 0(I n /I),X)\Hom(\pi_0(I_n^{\bullet/I}),X).

But it is easy to understand the combinatorics of the connected components of I n /II_n^{\bullet/I}, and hence compute this limit to be XX itself. This shows that the map XHom(I,X)X\to \Hom(I,X) is an isomorphism for truncated anima XX, but then also for all XX via a Postnikov limit.

What have we proved? That for all disctete anima XX, the map XHom(I,X)X\to \Hom(I,X) is an isomorphism of anima, where the right-hand side is the (external) Hom in condensed anima. But in fact, it is an isomorphism of condensed anima (i.e., is also true when using the internal Hom), and reading closely, the whole proof works internally. The one thing has to check is that discrete anima XX satisfy the equation X(S)=colim iX(S i)X(S)=\colim_i X(S_i), i.e. Hom(S,X)=colim iHom(S i,X)\Hom(S,X) = \colim_i \Hom(S_i,X), internally, which is again true.

Posted by: Peter Scholze on April 23, 2020 10:11 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Right, I was too blithe about the passage to the inverse limit, since there aren’t natural transition maps between the pushout squares I was considering! Thanks for the correction, Mike and Peter, and thanks, Peter, for giving a correct argument :)

Posted by: Dustin Clausen on April 23, 2020 11:52 AM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

No worries :-).

Maybe contemplating this proof is actually a good first exercise in understanding the condensed perspective. Note two ingredients:

– infinitary constructions, namely the passage to the limit I I_\infty, are absolutely critical. At each finite level, one would simply reduce contractibility of II to contractibility of the 2 n2^n smaller intervals in I nI_n. But in the limit, one actually resolves the problem.

– The existence of projective objects. This is used to see that sequential limits of surjections are still surjections (Bhatt and I call such topoi replete; there are some cases where one does not have projectives, but that’s the best situation), and hence is what allows this infinitary construction. (By the way, things come full circle here in a way, as the original motivation for Bhatt and myself in introducing the pro-etale topos was to make the sheaf =lim n/ n\mathbb{Z}_\ell = \lim_n \mathbb{Z}/\ell^n well-behaved.)

Geometrically, it is a reduction of seemingly nice ([0,1][0,1]) to seemingly pathological (the Cantor set I I_\infty) spaces.

Posted by: Peter Scholze on April 23, 2020 1:46 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Yes, good points! Let me add a small comment. In the argument we’re dealing with surjections between coherent objects, and there no countability assumption is required: an arbitrary limit of surjections is a surjections. Considering fibers, this just reduces to the standard fact that a filtered inverse limit of non-empty compact hausdorff spaces is non-empty.

Posted by: Dustin Clausen on April 23, 2020 3:21 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Sorry, I hope it was clear from the next sentence that I meant an arbitrary filtered limit!

Posted by: Dustin Clausen on April 23, 2020 4:18 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Dear Prof. Scholze and Prof. Clausen,

I learn analysis on metric spaces, which is rather distant from the algebraic/categorical world. Naturally I do not understand much about your notes on condensed mathematics and analytic geometry. Sorry if the following are just non-sense.

Your idea of resolving real numbers using profinite structures seems related to the so called dyadic cube systems on metric (measure) spaces. The basic idea is to construct a scale-depending system of partitions for a compact (or noncompact and doubling) metric space, in a simliar way as the natural dyadic cube system on Euclidean spaces, as done in classical harmonic analysis. It has a lot of applications in harmonic analysis, geometry of Banach spaces, analysis on fractals and even theoretical computer science.

I first learned this line of research from Naor’s survey “an introduction to the Ribe program”. The basic philosophy is that “ultrametric spaces are ubiquitous”, quoting Naor. Namely, metric spaces can be nicely approximated by (random) ultrametric ones. Lee and Naor applied the so called random partition technique to metric embedding problems, and Lee etc. found applications to spectral analysis of graphs, and to theoretical computer science.

In harmonic analysis, there were also intensive studies. Using random dyadic system, Nazarov, Treil and Volberg extended a lot of results in classical harmonic analysis to the non-homogeneous setting. Tolsa solved Vitushkin’s semi-additivity conjecture and Hytonen solved the A2 conjecture.

There’s also the recent work of Kigami. His idea is to view a compact metrizable space as being resolved by an augmented tree. Metrics and measures on the space are encoded in a unified way, as weight functions on the tree. Important types of relations between two metrics (or a metric and a measure) can be characterized as certain relations between corresponding weight functions. It has a flavor of “functors”.

I’m aware that the above results are analytic in nature and do not involve algebraic structures. Still I’m really curious whether there’s some connection. Is your resolution of real numbers secretly doing some harmonic analysis at the same time?

Best regards, Xueping

Posted by: Xueping Huang on July 22, 2020 3:36 PM | Permalink | Reply to this

Re: Pyknoticity versus Cohesiveness

Does the microcosm principle work for large Lawvere theories?

We have pyknotic sets as models of a large Lawvere theory here.

Posted by: David Corfield on April 28, 2020 11:29 AM | Permalink | Reply to this

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