Pyknotic set: Difference between revisions
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{{Short description|Mathematical concept}} | {{Short description|Mathematical concept}} | ||
In mathematics, especially in [[Topology|topology]], a '''pyknotic set''' is a | In mathematics, especially in [[Topology|topology]], a '''pyknotic set''' is a sheaf of sets on the [[Grothendieck topology|site]] of compact [[Hausdorff space]]s (with some fixed [[Grothendieck universe]]s). The notion was introduced by [[Biography:Clark Barwick|Barwick]] and Haine to provide a convenient setting for [[Homological algebra|homological algebra]].<ref>{{harvnb|Barwick|Haine|2019}}</ref> The term ''pyknotic'' comes from the Greek πυκνός, meaning dense, compact or thick.<ref>{{harvnb|Barwick|Haine|2019|loc=§ 0.1}}</ref> The notion can be compared to other approaches of introducing [[Generalized space|generalized space]]s for the purpose of homological algebra such as Clausen and [[Biography:Peter Scholze|Scholze]]'s condensed sets or Johnstone's topological topos.<ref>{{Cite web |title=Condensed vs pyknotic vs consequential |url=https://mathoverflow.net/questions/441838/condensed-vs-pyknotic-vs-consequential/ |access-date=2024-07-10 |website=MathOverflow |language=en}}</ref> | ||
Pyknotic sets form a [[Coherent topos|coherent topos]], while condensed sets do not.<ref>{{harvnb|Barwick|Haine|2019|loc=§ 0.3}}</ref> Comparing pyknotic sets with his approach with Clausen, Scholze writes:<ref>{{harvnb|Scholze|2019|loc=p. 7}}</ref> | Pyknotic sets form a [[Coherent topos|coherent topos]], while condensed sets do not.<ref>{{harvnb|Barwick|Haine|2019|loc=§ 0.3}}</ref> Comparing pyknotic sets with his approach with Clausen, Scholze writes:<ref>{{harvnb|Scholze|2019|loc=p. 7}}</ref> | ||
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* https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html | * https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html | ||
[[Category: | [[Category:Sheaf theory]] | ||
[[Category:Homological algebra]] | |||
{{Sourceattribution|Pyknotic set}} | {{Sourceattribution|Pyknotic set}} | ||
Latest revision as of 20:24, 14 April 2026
In mathematics, especially in topology, a pyknotic set is a sheaf of sets on the site of compact Hausdorff spaces (with some fixed Grothendieck universes). The notion was introduced by Barwick and Haine to provide a convenient setting for homological algebra.[1] The term pyknotic comes from the Greek πυκνός, meaning dense, compact or thick.[2] The notion can be compared to other approaches of introducing generalized spaces for the purpose of homological algebra such as Clausen and Scholze's condensed sets or Johnstone's topological topos.[3]
Pyknotic sets form a coherent topos, while condensed sets do not.[4] Comparing pyknotic sets with his approach with Clausen, Scholze writes:[5]
In a recent preprint [BH19], Barwick and Haine set up closely related foundations, but using different set-theoretic conventions. In particular, they assume the existence of universes, fixing in particular a “tiny” and a “small” universe, and look at sheaves on tiny profinite sets with values in small sets; they term these pyknotic sets. In our language, placing ourselves in the small universe, this would be κ-condensed sets for the first strongly inaccessible cardinal κ they consider (the one giving rise to the tiny universe).
References
Sources
- Barwick, Clark; Haine, Peter (2019). "Pyknotic objects, I. Basic notions". arXiv:1904.09966 [math.AG].
- Scholze, Peter (2019). "Lectures on Condensed Mathematics". https://www.math.uni-bonn.de/people/scholze/Condensed.pdf.
- Wolf, Sebastian (2020). "The Pro-Étale Topos as a Category of Pyknotic Presheaves". arXiv:2012.10502 [math.AG].
External links
- https://ncatlab.org/nlab/show/pyknotic+set
- https://mathoverflow.net/questions/441610/properties-of-pyknotic-sets
- https://mathoverflow.net/questions/356618/what-is-the-precise-relationship-between-pyknoticity-and-cohesiveness
- https://golem.ph.utexas.edu/category/2020/03/pyknoticity_versus_cohesivenes.html
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