Condensed Sets and the Solovay Model

We exhibit a geometric morphism from the Grothendieck topos representing the Solovay model to the $\kappa$-pyknotic sets of Barwick–Haine and Clausen–Scholze. We then use the properties of this morphism and automatic continuity in the Solovay model to prove Clausen–Scholze’s resolution of the Whitehead problem for discrete condensed abelian groups. We also exhibit an analogous internal $Ext$ computation between locally compact abelian groups in the Solovay model.