Condensed Sets and the Solovay Model

TL;DR

This paper constructs a geometric morphism linking the Solovay model to pyknotic sets, enabling the proof of the Whitehead problem for condensed abelian groups and computing internal Ext groups within the model.

Contribution

It introduces a new geometric morphism connecting the Solovay model with pyknotic sets and applies it to solve the Whitehead problem and compute Ext groups in this setting.

Findings

Proof of Clausen--Scholze's Whitehead problem resolution for condensed abelian groups

Construction of a geometric morphism from the Solovay model to pyknotic sets

Internal Ext computation for locally compact abelian groups in the Solovay model

Abstract

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.