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).

• Barwick, Clark; Haine, Peter (2019). "Pyknotic objects, I. Basic notions". arXiv:1904.09966 [math.AG].
• Scholze, Peter (2019). "Lectures on Condensed Mathematics" (PDF). Department of Mathematics, University of Bonn. Retrieved 2024-09-16.
• Wolf, Sebastian (2020). "The Pro-Étale Topos as a Category of Pyknotic Presheaves". arXiv:2012.10502 [math.AG].

• 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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