pyknotic set

{{Short description|Mathematical concept}}

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.{{harvnb|Barwick|Haine|2019}} The term pyknotic comes from the Greek πυκνός, meaning dense, compact or thick.{{harvnb|Barwick|Haine|2019|loc=§ 0.1}} 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.{{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}}

Pyknotic sets form a coherent topos, while condensed sets do not.{{harvnb|Barwick|Haine|2019|loc=§ 0.3}} Comparing pyknotic sets with his approach with Clausen, Scholze writes:{{harvnb|Scholze|2019|loc=p. 7}}

{{blockquote| 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).}}