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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeMay 1st 2019

Page created, but author did not leave any comments.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeMay 5th 2019

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeJun 11th 2019

Added a comparison made to condensed sets as not forming a topos.

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeJun 11th 2019

Added that the topos of pyknotic sets is not cohesive.

• CommentRowNumber5.
• CommentAuthorDavid_Corfield
• CommentTimeJun 11th 2019

Added a few more bits and pieces.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 11th 2019

Have not been following any details, but just glancing at the entry right now, something needs qualification here: Presently it says that condensed sets do not form a topos, but following the link gives that condensed sets are sheaves on some site. Something amiss here. I guess some size issues?

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeJun 11th 2019

Oh yes, I remember thinking that was odd.

As emphasised by Scholze, however, the distinction between pyknotic and condensed does have some consequences beyond philosophical matters. For example, the indiscrete topological space $\{0,1\}$, viewed as a sheaf on the site of compacta, is pyknotic but not condensed (relative to any universe). By allowing the presence of such pathological objects into the category of pyknotic sets, we guarantee that it is a topos, which is not true for the category of condensed sets. (p. 4)

But Scholtze speaks of the topos of condensed sets. It sounds like size is at stake. What’s going on exactly?

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeJun 11th 2019

It sounds like it’s just size technicalities.

• CommentRowNumber9.
• CommentAuthorDavid_Corfield
• CommentTimeJun 11th 2019
• (edited Jun 11th 2019)

Seems likely. The paragraph before the one I cited in #7, says of the difference between approaches that ” it is a matter of set theory”.

But still, what should be written on our pages when one person says condensed sets form a topos and another says they don’t?

• CommentRowNumber10.
• CommentAuthorMike Shulman
• CommentTimeJun 11th 2019

In the other discussion I just linked to, it sounded to me as though condensed sets were an elementary topos but not a Grothendieck one. But I have not actually read any of the literature myself.

Could we include on the page some explanation of the origin of the bizarre word “pyknotic”, whatever it might be?

• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeJun 11th 2019

• CommentRowNumber12.
• CommentAuthorDavid_Corfield
• CommentTimeJun 11th 2019

For any uncountable strong limit cardinal $\kappa$, the category of $\kappa$-condensed sets is the category of sheaves on the site of profinite sets of cardinality less than $\kappa$, with finite jointly surjective families of maps as covers.

The category of condensed sets is the (large) colimit of the category of $\kappa$-condensed sets along the filtered poset of all uncountable strong limit cardinals $\kappa$.

• CommentRowNumber13.
• CommentAuthorMike Shulman
• CommentTimeJun 11th 2019

Thanks. I wonder whether it is related to the infinitary-pretopos of small sheaves on the large site of all profinite sets.

• CommentRowNumber14.
• CommentAuthorDavid_Corfield
• CommentTimeMar 5th 2020

Added a talk given by Clark Barwick, apparently 1 of 4. Twitter conversation then provided more definitions, so I’ve added these.

• CommentRowNumber15.
• CommentAuthorDavidRoberts
• CommentTimeMar 5th 2020

• CommentRowNumber16.
• CommentAuthorDavid_Corfield
• CommentTimeMar 5th 2020

I’m not sure there’s anything of permanent value there, but it start with Emily Riehl here.

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeMar 5th 2020

It would be good to have a better sense of why they’re deliberately avoiding cohesion - something to do with a mismatch of topology and algebra.

I suspect there’s some connection to Lurie’s interest in Makkai, as here. Perhaps the Awodey school alternative to Makkai is relevant. All very vague, I know.

• CommentRowNumber18.
• CommentAuthorDavid_Corfield
• CommentTimeMar 6th 2020

Since I was following the first of Barwick’s lectures I thought I’d jot down some of the motivation he gives. Haven’t finished yet.

• CommentRowNumber19.
• CommentAuthorTim_Porter
• CommentTimeMar 6th 2020

There is a page: https://shifted-project.github.io/ on his SHIFTED project, that may be relevant. I have not yet tackled the pyknotic paper, but have glanced at some of the others.

• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeMar 6th 2020

SHIFTED: Stratified Homotopy Invariants, Field Theories, Exodromy, and Duality

Thank goodness it wasn’t ’Actions’ rather than ’Invariants’.

• CommentRowNumber21.
• CommentAuthorDavid_Corfield
• CommentTimeMar 6th 2020

There’s some connection between what they’re doing in BarHai19, sec 4.3 and Lurie’s work on Makkai’s conceptual completeness, as mentioned here.

• CommentRowNumber22.
• CommentAuthorSam Staton
• CommentTimeMar 7th 2020
Mike cited quasi-topological spaces, which do seem relevant. But do the pyknoticians ever reference or compare to these explicitly?
• CommentRowNumber23.
• CommentAuthorDavidRoberts
• CommentTimeOct 26th 2021

Condensed sets were only described as “not forming a topos”, whereas it’s more constructive to say what they actually do form: an infinitary pretopos.