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.
[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 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).