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

  1. ^ Barwick & Haine 2019
  2. ^ Barwick & Haine 2019, § 0.1
  3. ^ "Condensed vs pyknotic vs consequential". MathOverflow. Retrieved 2024-07-10.
  4. ^ Barwick & Haine 2019, § 0.3
  5. ^ Scholze 2019, p. 7

Sources


Allikas: https://en.wikipedia.org/wiki/Pyknotic_object
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