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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”.
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