Something that usefully emerged for me from this discussion and follow-up MO question is that rather than see cohesiveness and condensedness/pyknoticity in rivalry with one another, as my initial title proposed, that the former should be understood as a relative notion, constructions relative to a choice of base $\infty$-topos, while the latter involves the choice of a specific such base. This would suggest that a useful thing to investigate is the replacement in all things cohesive of the default base $\infty$-topos of $\infty$-groupoids by $Sh_{\infty}(ProFinSet)$. We might expect then that the package of constructions of cohesion in its various forms (infinitesimal, differential, singular (for orbifolds), elastic, solid, etc.) would be automatically available for this new base.

What I would like to gain a better sense of is how important such constructions are likely to be.

As a starter, then, have people considered the tangent $\infty$-topos, $T(Sh_{\infty}(ProFinSet))$, a case of infinitesimal cohesion over the new base? Presumably this involves some form of parameterised pyknotic/condensed spectra.

From the above-mentioned conversations, it appears that the $\infty$-topos of $\infty$-sheaves over the pro-étale site on all schemes over a separably closed field $k$ is cohesive over $Sh_{\infty}(ProFinSet)$. Then for any spectrum object in such a cohesive setting, the associated differential cohomology follows from the relevant differential cohomology hexagon. Is this likely to be important?

The article by Hisham Sati & Urs Schreiber, Proper Orbifold Cohomology, covers all forms of cohesion.


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Let me try to cut through the jargon. One thing that confuses me are two uses of "tangent spaces" here, that I believe are quite unrelated. One is the usual notion of tangent spaces of smooth manifolds say, based on which one defines differentials, and all sorts of de Rham cohomology etc.; I believe the "differential cohomology" is of this sort. On the other hand, there is the notion of the "tangent $\infty$-topos". I vaguely understand the reason for also calling this "tangent", but it is by a series of analogies, and there seems to be no relation between these two concepts in the case at hand.

More specifically, for condensed anima, the tangent $\infty$-topos is simply the $\infty$-category of pairs $(X,A)$ where $X$ is a condensed anima and $A\in \mathcal D(\mathrm{Cond}_{/X},\mathbb S)$ is a hypercomplete sheaf of spectra on the site of condensed sets over $X$. This is definitely a very interesting structure. This whole concept of $6$-functor formalisms is very much about such categories. Working with torsion coefficients, and allowing only "relatively discrete" coefficients, I've developed something along those lines in Etale cohomology of diamonds. Something even closer is in Chapter VII of Geometrization of the local Langlands correspondence (link should be active in a few days is active), where we restrict to the solid objects in $\mathcal D(\mathrm{Cond}_{/X},\mathbb Z_\ell)$. A critical role is then played by the left adjoint $f_\natural$ to pullback $f^\ast$. These do not exist in any classical setup, but have excellent formal properties. In fact, one gets a variant of a $6$-functor formalism where homology and cohomology are now on equal footing again (and arguably homology is even more primitive, again): The pullback functor $f^\ast$ admits a left adjoint $f_\natural$ ("homology") and a right adjoint $Rf_\ast$ ("cohomology"), both of which commute with any pullback. Moreover, Poincare duality holds for proper smooth maps. So yes, there's something interesting about this.

On the other hand, the reference to the differential cohomology hexagon in the question confuses me. For condensed sets, there are no tangent spaces, no differential forms, etc., and you can't get them back by magic. I think the problem is that I have no idea what the term "differential cohesion" means, but my strong feeling is that to have "differential cohesion" one needs extra structure like tangent spaces on the model spaces (and that passing to the "tangent $\infty$-topos" is not at all supplying these, as this is a very different procedure).


Edit: Thanks to the David's for their enlightening comments! Now I understand that the part with the differentials is really only in the examples, not in the "differential cohomology hexagon". For my convenience, let me reformulate this hexagon in my own language.

Say $X$ is a condensed anima, and $A\in \mathcal D(\mathrm{Cond}_{/X},\mathbb S)$ is a sheaf of spectra on condensed sets over $X$. (Or take $X$ a scheme, and $A\in \mathcal D(X_{\mathrm{proét}},\mathbb Z_\ell)$, or $X$ a small v-stack and $A\in \mathcal D(X_v,\mathbb Z_\ell)$, or...) Let $\pi$ denote the projection from the site of $X$ to the (pro-étale) site of the point. Then pullback $\pi^\ast$ has a right adjoint $R\pi_\ast$ ("cohomology") and a left adjoint that I will denote $\pi_\natural$ ("homology").

In particular, we get condensed spectra $R\pi_\ast A$ (=$R\Gamma(X,A)$), the cohomology of $A$, and $\pi_\natural A$, the homology of $A$. By adjunction, we get maps $$ \pi^\ast R\pi_\ast A\to A\to \pi^\ast \pi_\natural A. $$ Let $\overline{A}=\mathrm{cofib}(\pi^\ast R\pi_\ast A\to A)$ and $\tilde{A}=\mathrm{fib}(A\to \pi^\ast \pi_\natural A)$. Then there is a pullback square $$\begin{CD} A @>>> \overline{A}\\@VVV @VVV\\ \pi^\ast \pi_\natural A @>>> \pi^\ast \pi_\natural \overline{A} \end{CD} $$ and a pushout square $$\begin{CD} \pi^\ast R\pi_\ast \tilde{A} @>>> \pi^\ast R\pi_\ast A\\@VVV @VVV\\ \tilde{A} @>>> A \end{CD} $$ Of course, pushout squares and pullback squares are equivalent, but I want to stress that one wants to use them to recover $A$ from "simpler" information. However, to me $\tilde{A}\to A\to \overline{A}$ all feel extremely similar, and I'd regard these squares as simple statements about how to analyze the small difference between them.

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    $\begingroup$ Thanks, Peter. To add to the thicket of tangent notions, there's the work of Ching et al.. I started a conversation about that here: golem.ph.utexas.edu/category/2021/02/…". The differential cohomology hexagon construction on the nLab is meant in a very general sense. Hence the warning "Beware that this is a very general conceptualization of de Rham coefficients". It just needs a cohesive ∞ -topos, and then the choice of a stable object in the latter. As far as I know, such resulting fracture squares should work for relative cohesion too. $\endgroup$ Feb 25, 2021 at 12:28
  • $\begingroup$ FWIW I think the only place "tangent spaces" could be hiding in the statement ncatlab.org/nlab/show/… would be in the "flat modality". Everything else there seems to me to be rather general cohesion-type technology (certainly the shape modality might be a bit special, but doesn't seem to be anything to do with tangent spaces). (Edit: cross-posted with David C) $\endgroup$
    – David Roberts
    Feb 25, 2021 at 12:31
  • $\begingroup$ I believe the construction works even when that left adjoint doesn't preserve finite products. So then choosing a stable object in some ∞ -topos which is relatively cohesive over condensed anima should allow such a fracture square, I think. $\endgroup$ Feb 25, 2021 at 12:31
  • $\begingroup$ Peter, you mentioned elsewhere that ∞-sheaves over the big pro-etale site on all schemes over a separably closed field k are relatively cohesive over condensed anima. What if you took a stable object in the former and formed the pullback square, using the corresponding Π and ♭dR? Does that provide a more interesting factorization? $\endgroup$ Feb 25, 2021 at 16:50
  • $\begingroup$ Ah, yes, you can also do that, it's implicit in the "etc." above. For the moment, I don't see any interesting examples of these squares, but I'll keep them in mind. $\endgroup$ Feb 25, 2021 at 19:28

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