Condensed Thursday

This is the Fall 2020 edition of the Thursday seminar. We will learn about Clausen-Scholze/Barwick-Haine’s theory of “condensed/pyknotic mathematics.”

Organizers: E.E., P. H


  1. Scholze’s lectures [S]
  2. Scholze’s sequel [S2]
  3. Barwick-Haine [BH]
  4. Exodromy [Ex]
  5. Hoffman-Spitzweck [HS]
  6. Seminar by Cisinski and Scheimbauer


  1. Grievances (Sept 10). (Elden and Peter).
  2. Condensed sets or “pro-étale sheaves on a point” (Sept 17). Introduce condensed sets and prove Proposition 1.7 of [S] that embeds the category of topological spaces into condensed sets. Time permitting, discuss the set-theoretic issues involved and/or the genesis of condensed sets via the pro-étale topology (Lucy).
  3. Condensed abelian groups (Sept 24). Define condensed abelian groups and prove Theorem 2.2 of [S]. (Shachar)
  4. Cohomology (Oct 1). Prove the comparison theorem, Theorem 3.2 of [S], between condensed cohomology and sheaf cohomology. (Piotr)
  5. LCA’s as condensed abelian groups and resolution theorem (Oct 8). Prove the Theorem 4.10 of [S] and give lots of examples of computations of RHom‘s between condensed abelian groups (Elden) (Notes).
  6. Solid abelian groups I (Oct 15). Explain solid abelian groups as “complete” condensed abelian group objects; this takes two lectures. Prove Theorem 5.4 of [S] and explain its corollaries (Martin/Jay).
  7. Solid abelian groups II (Oct 22). Prove Theorem 5.8 of [S]. (Martin/Jay)
  8. Analytic rings and solid modules (Oct 29). Motivate the theory of analytic rings by reviewing some theory of affinoids. Explain what analytic rings and modules are. Explain how the left adjoint to the left adjoint of restriction exists (!), Theorem 8.1 of [S] (Peter).
  9. Q&A (Nov 5) (condensed questions)
  10. Globalization I (Nov 12). Review Huber’s formalism of analytic geometry and show how this is recovered by stating Theorem 9.8 of [S] (Dori/Sanath).
  11. Globalization II (Nov 19). Prove Theorem 9.8 of [S] (Dori/Sanath).
  12. Grothendieck duality (Dec 3). Prove coherent duality, Theorem 11.1 of [S] (Dylan).
  13. Condensed mathematics and exodromy (Dec 10). (Lucy)