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Let $\mathcal E$ be an $\infty$-topos. Recall that Lurie defines the shape of $\mathcal E$ as the left-exact, accessible functor $\Gamma \Delta: Spaces \to Spaces$ where $\Delta: Spaces^\to_\leftarrow \mathcal E: \Gamma$ is the the constant / global sections adjunction, the unique geometric morphism from $\mathcal E \to Spaces$. The $\infty$-category of left-exact, accessible endofunctors of $Spaces$ is better-known as the $\infty$-category of Pro-spaces.

As a pro-space, the shape of $\mathcal E$ is like having a space, but even better, because its homotopy groups are not just groups, but pro-discrete groups. For instance, the shape of the etale site of a scheme $X$ is essentially the etale homotopy type of $X$, and its fundamental groupoid is essentially the etale fundamental groupoid, complete with its usual profinite topology (I am glossing over some additional finiteness considerations; to be honest this is largely because I don't fully understand them and have even less idea how they change the picture conceptually.)

This situation -- of homotopy groups which come with a topology -- is (from some perspective) exactly what the condensed mathematics of Clausen and Scholze / pyknotic mathematics of Barwick and Haine was developed for, and my understanding is that the example of the etale homotopy type is particularly germane, as Bhatt and Scholze's first application of the theory is to study the pro-etale site and its associated fundamental group. So I think the answer to the following questions must be yes, at least modulo the finiteness conditions I haven't understood:

Question 1: Is there a condensed / pyknotic "re-interpretation" of the pro-discrete topology on the homotopy groups of the shape of an arbitrary $\infty$-topos?

Question 2: Is there a condensed / pyknotic refinement of the shape of an arbitrary $\infty$-topos, perhaps containing more information than the usual pro-space?

I think if I read Bhatt and Scholze, it should all be in there. Unfortunately, I'm hampered by the fact that the etale fundamental group is usually approached differently by algebraic geometers, in a way which circumvents considering all of the data of the etale homotopy type. This approach is doubtless more practical for algebraic geometry (and of course, my way of putting it is completely ahistorical, as Grothendieck constructed the etale fundamental group long before Artin and Mazur constructed the etale homotopy type), but unfortunately I am not an algebraic geometer and I've never dug into it to understand. It appears that Bhatt and Scholze use a similar approach, which is an obstacle for me to understand it. So I'm wondering if somebody can translate these formal aspects of the situation into a language which I personally happen to understand better.

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    $\begingroup$ Side question: the etale homotopy type (Artin-Mazur) recovers the SGA3 (pro-discrete) etale fundamental group, which is smaller than the pro-etale fundamental group (Bhatt-Scholze). Do we expect the refinement you seek to have the pro etale fundamental group as its $\pi_1$? It’s not always pro-discrete as a group, only as a topological space (more precisely, a Noohi group). $\endgroup$
    – Piotr Achinger
    Commented Feb 8, 2021 at 8:12
  • $\begingroup$ @PiotrAchinger Good question. Clearly I haven't considered this carefully, but it's already a bit of a lie to say that we're working with pro-discrete spaces, since unlike in the profinite case, the "take the limit" functor $Pro(Set) \to Top$ is not full (e.g. a pro-system of infinite sets can have empty limit without being pro-isomorphic to the empty set). So the future work of Barwick, Glasman, and Haine addressing this conjecture has a few conceptual stumbling blocks to work out! $\endgroup$
    – Tim Campion
    Commented Feb 8, 2021 at 16:26

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In subsection 13.8.10. of version 7 of the "Exodromy" paper by Barwick, Glasman and Haine (arXiv:1807.03281v7), the authors define the "pyknotic étale homotopy type". They say that it should recover the pro-étale fundamental group of Bhatt–Scholze and will explore this in a future work.

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    $\begingroup$ See also Sebastian Wolf's beautiful paper arxiv.org/abs/2012.10502. $\endgroup$
    – Clark Barwick
    Commented Feb 8, 2021 at 11:25
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    $\begingroup$ This certainly validates the idea that the pro-etale fundamental group is indeed the pyknotic fundamental group of a pyknotic space, but hang on -- the construction described in 13.8.10 is the pyknotic analog of the geometric realization of a category -- not the shape of an $\infty$-topos. (This actually makes it potentially even more interesting to me for unrelated reasons...) I ought to know the relation between geometric realization and shape in ordinary $\infty$-categories, but I don't... $\endgroup$
    – Tim Campion
    Commented Feb 8, 2021 at 16:32
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    $\begingroup$ @TimCampion : A naive comment as I would need to spend much more time reading this paper to understand all the notation. But an obvious connection between shape and realization is that the shape of Prsh(C) is an infinity groupoid, and corresponds to the realization of the category C. So given that what they call Gal(X) is (if I understand correctly) a topologized version category of points, that does makes a lot of sense. $\endgroup$
    – Simon Henry
    Commented Feb 8, 2021 at 16:38
  • $\begingroup$ @SimonHenry Thanks, I should have remembered / realized that. In fact, it would make perfect sense if the pro-etale topos is a pyknotic presheaf topos. It's still mysterious to me that I don't see how the topology one gets from being a pro-system relates to the topology that one gets from being pyknotic... $\endgroup$
    – Tim Campion
    Commented Feb 8, 2021 at 16:43
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    $\begingroup$ @TimCampion The "obvious" way is to take the limit of the pro-object in the category of condensed objects (this should be lossy). Maybe it is helpful to look at the comments in this answer. I believe him that in general, pro-objects don't seem to be well-behaved, and condensed objects are much better behaved. $\endgroup$
    – user20948
    Commented Feb 13, 2021 at 17:50

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