### 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 $(\infty, 1)$-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 ‘$p$-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 $Pyk$ construction applies to all finite-product categories, $D$. There are several equivalent formulations of the concept, one being that $Pyk(D)$ is composed of the finite-product-preserving functors from the category of complete Boolean algebras to $D$. (See others sites at pyknotic set.) We thus have $Pyk(Ab)$, the category of pyknotic abelian groups.

One reason, we are told, for the whole pyknotic approach is that $Pyk(Ab)$ rectifies a perceived problem with the category of topological abelian groups, $AbTop$, 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 $AbTop$ 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.]

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

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 $CompBoolAlg$ 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.

## 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.