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hyperlinked homotopical alggebra and, notably, pro-étale site
the natural map
$T(S)\to T(S')\rightrightarrows T(S'\times_S S')$is a bijection
Which map is a bijection?
It’s a fork, not a map. Fixed now.
Thanks. I wonder if we can understand this development of condensed mathematics via cohesion. So there are situations where we are looking to do algebra with objects with a topology, but a unified method is lacking to deal with all such situations. Condensed mathematics is proposed as a solution.
On p.6 some examples of problems are given. The first is that AbTop, the category of topological abelian groups, is not itself an abelian category. For instance we have a map from $\mathbb{R}$ with the discrete topology to $\mathbb{R}$ with the natural topology. However,
In an abelian category, the failure of this map to be an isomorphism has to be explained by a nontrivial kernel or cokernel.
Then the category of condensed abelian groups is abelian. (p. 11)
So does this problem not arise in the cohesive treatment? For the map $\flat \mathbb{R}$ to $\mathbb{R}$ there is the homotopy fiber of the suspension of this counit, as at differential cohomology diagram, designed particularly for stable objects.
The second problem is
For a topological group $G$, a short exact sequence of continuous $G$-modules does not in general give long exact sequences of continuous group cohomology groups. More abstractly, the theory of derived categories does not mix well with topological structures.
Does cohesion say something here?
In view of the rival pyknotic sets, I guess we can ask similar questions. There’s a right and left adjoint to the underlying space of a pyknotic set.
Barwick and Haines explicitly point out that there’s no further left adjoint, so not cohesive. I should add that over there.
I was wondering the same when I saw this development, but haven’t been following any details. On general grounds, cohesion or not, we have that mixing topology into group theory and homological algebra is of course the topic of stacks and higher topos theory. Here apparently mostly over a pro-etale site. Some months back at the modal type theory meeting in Pittsburgh, there was one coffeee break that saw some people get optimistic again about pro-etale toposes being cohesive. Would be a major thing, I hope somebody looks into it.
The failure of $TopAb$ to be an abelian category is the same as the failure of $Top$ to be balanced, and has the same solution: take cohesion to be stuff rather than structure in general. In particular, the category of abelian group objects in any topos is abelian; in a cohesive setting (and probably more generally too) the map $\flat\mathbb{R} \to \mathbb{R}$ is injective but not surjective, and its cokernel is a nontrivial abelian group with (presumably) “only one point” but more “cohesive stuff”. This must also be what’s going on with condensed sets and pyknotic sets.
I’d like to understand this better. Barwick and Haines set things up so as deliberately to avoid cohesion:
the topos $\mathbf{Pyk}(\mathbf{S})$ is – by design – not cohesive in the sense of Schreiber
Are you saying in #8, Mike, that the motivation to do so will disappear if cohesion is properly understood as stuff?
I can’t say, since I don’t know what their motivation is for making it not cohesive. That remark doesn’t say.
Well I guess it derives from this proétale business.
There is a deep connection between the passage from objects to pyknotic objects and the passage from the étale topology to the proétale topology. (p. 2)
So maybe one question is about how proétale-ness and cohesion fit together, in particular the discrete functor not have a left adjoint.
There are also links to related matters I’ve wondered about. In Sec 4.3 they look at ultracategories as studied by Lurie. He used them when taking up Makkai’s work on conceptual completeness. You’ll see in this MO comment that something proétale is in the air there.
[Note to self, if I ever have time, to look at Definability, interpretations and étale fundamental groups.]
Right, cohesion depends on local connectedness/contractibility of the model spaces. If the spaces people are interested in are not locally connected, then we shouldn’t expect a shape functor (the example I’m more familiar with is Johnstone’s topological topos) – the type theory then is usually what I called “spatial type theory” with $\flat$ and $\sharp$ but no shape. I believe that’s unrelated to the issue of balancedness in #8.
I wonder what the optimistic coffee break people of #7 who hope that pro-etale toposes are cohesive were thinking.
Any such people might like to look through a discussion between Urs and Marc Hoyois (especially from #12 onwards).
In my travels, I see that we were to merge proadjoint and pro-left adjoint, but never did. Is there a preference as to which is the main name and which the redirect?
Why is the section at the latter called Pro-étale homotopy type, but it only goes on to speak of étale homotopy types?
Thanks for reminding me. There is a magnificent PhD thesis topic waiting here, ready to be picked. Who will do it?
Perhaps that should be étale pro-homotopy type?
I was trying to figure out how the different ’pro-’s relate. So I take it the pro- in proadjoint is due the appearance of pro-objects. But I never wondered why the pro- in profunctor. No explanation is given there, but do we see from pro-left adjoint signs of a connection between these in that there is a profunctor factoring through pro-objects?
@David C since profunctors were also given names like distributors, bimodules etc, they aren’t obviously related to pro-objects.
The only choice of terminology worse then “profunctor” is “anafunctor”.
I started out closer to the other end ’obviously unrelated’. And that seems right from Mike’s comment:
Am I the only one who minds the clash of ‘profunctor’ with pro-object?
Yes, “profunctor” is unrelated to “pro-object”. Unfortunately I still prefer “profunctor” to “distributor” and “bimodule”.
I think I missed Mike’s comment of June 15. I agree about “distributor”, but I can’t understand what’s wrong with “bimodule”. I assume, Mike, you don’t have a problem with the name for the case of one-object $Ab$-categories. The general concept being so obviously useful, why not expand the semantic scope of the name?
I think that for me the generalization is too different to deserve keeping the same name. The sort of things that one does with profunctors, at least in pure category theory, are too different from the sort of things that one does with bimodules. In addition, there’s a potential for confusion: if $C$ and $D$ are monoidal categories, then does a “$C$-$D$-bimodule” mean a profunctor, or a category with monoidal actions of $C$ and $D$ on both sides?
The sort of things that one does with profunctors, at least in pure category theory, are too different from the sort of things that one does with bimodules.
That’s not my experience; can you give an example?
For example, defining limits and colimits in equipment-theoretic language: the limit of an arrow weighted by a proarrow.
I’d have a strong preference against bimodule, seems like an unnecessary conflation of concepts. Not sure really what is wrong with distributor: it doesn’t suggest anything in particular to me, but nor does it clash with anything, so it is at least harmless.
Conflation?? Tell it to Ross Street.
Hehe, I simply mean that a bimodule in the usual sense is not the same as a profunctor/distributor, no matter that there are some ways in which they are related, and I don’t think it’s helpful use exactly the same word in that case. It may be erudite, but not really helpful in general. Just my opinion though!
I don’t actually have a problem with profunctor (I’d probably write pro-functor if it were some kind pro object, with slightly more emphasis on the pro orally, and I don’t actually personally think of pro-objects when I read about profunctors), but it is true that there is evident potential for confusion!
no matter that there are some ways in which they are related
Not just “related”. The commutative algebra sense is a special case of the Australian category theory sense.
but not really helpful in general
It has been extremely helpful to me, and I daresay to many others.
“Profunctor” just doesn’t suggest what the concept is about. Neither does “distributor” (which I find worse: positively misleading). “Bimodule” at least serves to remind that we’re talking about a situation in which we have two monoid-like objects acting on either side of another object in a compatible way. People should wake up to the fact that this notion is extremely widespread, and not limited to rings.
In my opinion, the categorical notion of bimodule is so important, and subsumes as it does the more limited commutative algebra sense, that it makes sense to me to play Bourbaki and give that name to the more important and encompassing notion. (At least among friends!) The commutative algebra sense is too limiting for it to have the name all to itself, and leave the categorical notion struggling for the dignity of a good name.
Bottom line: I would (and will) use “bimodule” and simply let context decide which kind is being discussed. Realistically, I don’t think it’s all that confusing (and to be fair neither is “profunctor”, which I’ve also used believe it or not, even though I find it less descriptive).
The commutative algebra sense is a special case of the Australian category theory sense.
Yes, this was part of what I meant by ’related’, and is one part of why I think that it is not a good name. Group conjugations are (essentially) a special case of natural transformations, but we do not call natural transformations ’conjugations’. I may after all wish to refer to profunctors/distributors/bimodules without suggesting that commutative algebra is in any relevant or needed for what I am doing, i.e. for me, it is much better to have an unsuggestive term with no agenda than a term with an agenda.
But none of this is important to me, just adding my two pennyworth!
Todd, when talking in an abstract context of, say, monads in a bicategory or double category and bimodules between them, I certainly do use the word bimodule. So in that sense, I’m willing to use the word even more generally, since this includes profunctors and also all sorts of other things. But I also find it helpful to have a word that is specific to the case of categories, since as I said that case has particularly distinct behavior and uses.
I think that “profunctor” does suggest something of what the concept is about, namely that it’s a sort of generalization of a functor: a profunctor from $C$ to $D$ is a functor from $C$ to the presheaf category $P D$.
What would you say in the case of monoidal categories to distinguish the profunctors from the monoidal-structure-bimodules?
Finally, here’s another, somewhat more precise, reason that I don’t think of profunctors as the right sort of “generalization” of bimodules to deserve the same word. If we think of the parameter of a notion of “profunctor” as, say, a base enriching category $V$, then there is no $V$ such that $V$-enriched profunctors coincide with bimodules in the classical commutative algebra sense. There is a $V$ such that $V$-enriched profunctors include classical bimodules, but that’s a different statement. Of course there are other ways to slice things too, such as by fixing two $V$-categories or by working in the generality of a bicategory, but the existence of this way of parametrizing is, I think, part of what makes me dislike that terminology.
Mike #33: thanks for explaining in more detail. I wouldn’t say you’ve converted me, but at least I understand a bit better what you are driving at.
As for your question: eh, I haven’t thought about it to give anything more than an off-the-cuff response. Monoidal bimodules? What would you call it?
Or “bi-actegory”, perhaps…
Given monoidal categories $C$ and $D$, I would talk about “profunctors from $C$ to $D$” and “$C$-$D$-bimodules”, the former meaning the categorical notion and the latter the monoidal one.
Unless you declare your usage in the preface, that would be very confusing.
I think there’s a somewhat interesting difference in how I think of mathematicai nomenclature and how Todd and I think Mike as well think of it. Todd and Mike I think feel that a good name is one that typically is suggestive to some degree, e.g. gives some an indication of the meaning of the objects one is naming. Whereas I feel that this is not of especial significance. One could call something more or less anything that does not clash with something else, and as long as enough people use it, the name will ’take on’ the meaning of the thing it is describing.
There are numerous examples of this. ’Quandle’ would be a good one: I have no idea where the name comes from, it is not in the least degree suggestive to me, but it has acquired/’taken on’ the meaning of the algebraic object it describes, and I think it is a perfectly nice name! ’Kei’ is another one in the same family: it has some meaning in Japanese, but I don’t remember what it is; in mathematics it has taken on the meaning of the gadget it describes, and it is a nice name! ’D-module’ would be another example: I have no idea what ’D-’ stands for if anything, but it is doesn’t matter, the entire word ’D-module’ has taken on the meaning of the thing it describes. ’Club’ would be an example in category theory.
For examples of a different kind, take things named after people: ’abelian group’, ’Brauer group’, …, all completely unsuggestive, but they acquire the meaning of the object they describe.
There are even examples of terminology that a priori is definitively misleading that takes on the meaning of the thing it describes: ’perverse sheaf’ being a classic example. It was used by a group of influential people, everybody else started using it, there is no name clash, so it doesn’t really matter what the conventional meanings of ’perverse’ and ’sheaf’ are, one just reads ’perverse sheaf’ as a whole.
Of course there is a certain aesthetics in name choice as well, some names just do not feel right aesthetically, but otherwise I think any term without a priori connotations will work fine if it is adopted by a community. Of course something that happens to be suggestive is fine as well if it is a new word; one after all needs to get some inspiration from somewhere. But re-using a term from elsewhere brings baggage; one is basically choosing a particular intuition/point of view to force on others forever more. I generally prefer terminology to not get in the way rather than shout out to me; I do actually kind of cringe when I read ’bimodule’ in papers of Street, for example :-).
I take your point. But we may all have to learn to cringe around each other. :-)
Absolutely, as I say, it doesn’t really matter to me ultimately, I just thought I’d elaborate a bit on my point of view in case it is of use/thought-provoking for somebody :-).
Richard: I actually agree that otherwise-meaningless names are often fine. (And in particular, I’m more inclined than some to name things after people, thereby both giving credit to originators and obtaining an otherwise-meaningless name that is free to take on the desired meaning.) The problem I see here is that “bimodule” is not a meaningless name; it already has a different meaning in mathematics.
Todd: It wouldn’t be confusing if people hadn’t appropriated the word “bimodule” to mean “profunctor” in the first place! But yes, of course, I would declare the meaning in the introduction if I ever actually used such terminology.
But yes, of course, I would declare the meaning in the introduction if I ever actually used such terminology.
That was never in doubt! :-) No hard feelings I hope.
No, of course no hard feelings.
Added
Not regarding your edit, but a thought on that MO comment which you are pointing to:
Where this says “infinitesimally close” one should look for a better word, maybe “asymptotically close”, not to clash with well-established terminology for yet other notions of space.
And where it says “arguably more interesting” one should instead say what it’s good for, as mathematical complexity is not an end in itself. (Imagine Newton thinking that $F = m a$ might be too simplistic a relation for an honorable mathematician to promote, while $F = m arctan(a)$ would arguably be more interesting…)
When contrasting with the topological and bornological toposes in this thread, Scholze stresses convergence along ultrafilters:
In some sense condensed sets are the version of Johnstone’s topos that replaces convergence along sequences by convergence along ultrafilters;
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
Does this suggest a kind of closeness?
Regarding the “arguably more interesting”, in that same thread, I cited a motivation of Scholze
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.
and pointed out that Lawvere was similarly motivated concerning his bornological topos.
There must be better and more concrete motivations, though? Is this still in attack of some outstanding conjecture in arithmetic geometry (or similar), or is this now more philosophical exploration of the mathematical landscape?
It looks like we can tune into a symposium today to find out.
If I were serious about finding out, I’d dig through the literature. But I admit I am just firing off comment here while really being occupied with something else. But if we (i.e. somebody reading here) could add more and more concrete motivations and applications to condensed set, that would be great. (I am not doubting they exist. But all the more would it be good to add them.)
I was hoping that the discussion I started would get at the category-theoretic rationale. There ought to be a reason why the choice of sheaves over compacta is a natural one.
Once you indicated it had surfaced how the base topos of condensed sets makes algebraic/arithmetic pro-étale toposes be, if not cohesive then at least locally $\infty$-connected (i.e. having a second left adjoint to their global section functor).
Checking now at your notes at condensed cohesion I don’t recognize any statement there; did it remain unclear after all?
Let me suggest to clean up that entry: The mentioning of tangent $\infty$-toposes and of the differential cohomology hexagon there seem to best be omitted for the time being, since it seems it just muddies the waters at this point to make the second (or third) step before the first.
To my mind, what this entry should say, if there is anything to be said, is that the pro-étale topos over a suitable base ring is a topos over condensed sets (if that’s the case), and then something about an extra left adjoint and possible further extra structure.
If this statement is not available for the time being, then it might be better to clear that entry for now, as its title raises expectations which are being disappointed.
Yes, that page is not in a good state. But in #54 I was talking about the n-Cafe discussion. I was hoping for more insight on the base topos, why ultrafilters crop up there, etc. My suspicion was it had to relate to the codensity monad account of ultrafilters.
But, yes, #55 needs addressing. At the very least a re-ordering of sections.
According to unpublished work by Alexander Campbell, the universal property of the category of condensed sets is that it is the infinitary-pretopos completion of the pretopos of compact Hausdorff spaces. Presumably it would be appropriate to wait until there is at least a preprint to mention this on the nLab page?
If you write a little paragraph explaining the claim a little, then this will be useful in any case.
Can we add in that compact Hausdorff spaces are the algebras of the codensity monad induced by the inclusion of finite sets in the category of sets to some larger universal story?
An earlier talk to the one in #57 appears here:
How nice is the category of condensed sets?
Alexander Campbell – 16 December 2020
The new theory of “condensed mathematics” being developed by Clausen and Scholze promises to make analytic geometry amenable to the powerful techniques of modern algebraic geometry. The basic objects of this theory are the “condensed sets”, which may be defined as the small sheaves on the large site of compact Hausdorff spaces with the coherent topology. (These are nearly the same as the pyknotic sets of Barwick and Haine, up to issues of size.) In this talk we shall study several categorical properties of the category of condensed sets (which properties are surely known to the experts). We shall prove that this category is a locally small, well-powered, locally cartesian closed infinitary-pretopos, that it is neither a Grothendieck topos nor an elementary topos – since it lacks both a small generator (indeed, it is not even total) and a subobject classifier – but that it does have a large generator of finitely presentable projectives, and hence is algebraically exact. We shall also discuss the relationship of Spanier’s quasi-topological spaces to condensed sets.
Thanks! If you could add to such announcements something like “and so I fixed it” (as you did), that would help readers.
It’s still a little mysterious. While #62 speaks of “no such” functor, the edit that goes with it rev 26 seems to have kept the statement except for promoting “$\kappa$” in “$\kappa$-condensed” from a side remark to the main clause.
I think “no such” without the bounds was meant. Scholze writes
Removing the cardinal bounds from Proposition 1.7 below results in some surprising twist, cf. Warning 2.14, so we are careful with the cardinal bounds here.
Warning 2.14 is
Warning 2.14. The natural functor from topological spaces to sheaves on the full pro-étale site ∗proét of a point does not land in condensed sets…
I wonder if there’s anything further on Campbell’s characterisation of condensed sets (#57)
the infinitary-pretopos completion of the pretopos of compact Hausdorff spaces
Even with this, it hardly explains the claim:
condensed sets are meant to be “the topos of spaces”.
Added:
Several different sites can be used to define condensed sets, and, more generally, condensed ∞-groupoids:
Stone spaces (compact Hasdorff totally disconnected spaces);
Stonean spaces (compact Hausdorff extremally disconnected spaces).
In all three cases, morphisms are given by continuous maps and covering families are given by finite families of jointly surjective continuous maps.
The equivalence of sites is established in Yamazaki \cite{Yamazaki}.
Added:
The equivalence of various sites for condensed sets is established in
We should establish the relationship between the topos of light condensed sets (and I can say that without complaining it’s not literally true!) and Johnstone’s topological topos. The latter is defined as canonical sheaves on the category with objects $\ast$ and $\mathbb{N}\cup \{\infty\}$, and this category is a subcategory of that of light profinite sets (i.e. profinite sets that are countable sequential limits of finite sets; this includes the standard Cantor set). Light condensed sets are sheaves on light profinite sets, but now with the coherent topology.
I’m not in a position to figure this out right now, but it would be nice if the ff embedding of sequential spaces into light condensed sets extended to the topological topos, hence also relating subsequential spaces to light condensed sets.
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