Condensed Sets and the Solovay Model
Nathaniel Bannister, Dianthe Basak
TL;DR
The paper constructs a geometric bridge between κ-pyknotic/condensed sets and the Solovay model $V(\mathbb{R})$ by embedding $Sh(\mathbf{Sol})$ into the condensed/pyknotic framework and identifying its double-negation subtopos with the Solovay model category $\mathbf{VD}$. This bridge enables transfer of automatic-continuity phenomena from the Solovay model to condensed- and pyknotic-abelian groups, yielding Clausen–Scholze’s resolution of the Whitehead problem for discrete condensed abelian groups and a parallel internal Ext computation between $\mathbb R$ and $\mathbb Z$ in $\mathbf{VD}$. A sequence of carefully controlled images of familiar spaces and groups under the transfer functor $\Lambda$ clarifies how classical topological and algebraic objects behave in $\mathbf{VD}$, and the Solovay-model choice principles underpin a splitting argument that forces the vanishing of $\underline{Ext}(\underline{\mathbb R}, \underline{\mathbb Z})$. The work both abstracts a robust methodology for interfacing set-theoretic forcing with topos-theoretic condensed mathematics and suggests concrete avenues for extending these techniques to broader condensed-analytic contexts and higher-cardinal regimes.
Abstract
We exhibit a geometric morphism from the Grothendieck topos representing the Solovay model to the $κ$-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.