In mathematics, especially in

topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, a pyknotic set is a

sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper ser ...

of sets on the site of compact Hausdorff spaces (with some fixed

Grothendieck universe In mathematics, a Grothendieck universe is a set ''U'' with the following properties: # If ''x'' is an element of ''U'' and if ''y'' is an element of ''x'', then ''y'' is also an element of ''U''. (''U'' is a transitive set.) # If ''x'' and ''y'' a ...

s). The notion was introduced by Barwick and Haine to provide a convenient setting for

homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...

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

References

* *Peter Scholze, Lectures on Condensed Mathematics, 201

*Sebastian Wolf, The pro-étale topos as a category of pyknotic presheaves, arxiv/2012.10502

References

Further reading