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Added a link to quasi-topological space.
Added a comparison made to condensed sets as not forming a topos.
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?
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?
It sounds like it’s just size technicalities.
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?
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?
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 $\kappa$.
Thanks. I wonder whether it is related to the infinitary-pretopos of small sheaves on the large site of all profinite sets.
Can you give a link to these Twitter conversations?
I’m not sure there’s anything of permanent value there, but it start with Emily Riehl here.
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.
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.
SHIFTED: Stratified Homotopy Invariants, Field Theories, Exodromy, and Duality
Thank goodness it wasn’t ’Actions’ rather than ’Invariants’.
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.
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