Pyknotic set explained

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. The term pyknotic comes from the Greek πυκνός, meaning dense, compact or thick. 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.^[1]

Pyknotic sets form a coherent topos, while condensed sets do not. Comparing pyknotic sets with his approach with Clausen, Scholze writes:

References

• Barwick . Clark . Haine . Peter . Pyknotic objects, I. Basic notions . 2019 . math.AG . 1904.09966 .
• Peter Scholze, Lectures on Condensed Mathematics, 2019 https://www.math.uni-bonn.de/people/scholze/Condensed.pdf
• 2012.10502 . Wolf . Sebastian . The Pro-Étale Topos as a Category of Pyknotic Presheaves . 2020 . 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

Notes and References

1. Web site: Condensed vs pyknotic vs consequential . 2024-07-10 . MathOverflow . en.