Condensed Whitehead Problem

Abstract

The Whitehead problem, which asks whether every abelian group $A$ satisfying ${\mathrm{Ext}}^1(A,\mathbb{Z})=0$ is free, is independent of ZFC. However, this problem has an analogue in condensed mathematics that can be answered affirmatively. In this note, we give a new proof that an abelian group $A$ of size $\kappa$ is free if and only if ${\underline{\mathrm{Ext}}}^1(\underline{A},\underline{\mathbb{Z}})(S_{\kappa })=0$, where $S_{\kappa }$ is the Stone space of the Boolean completion of $\mathrm{Add}(\omega ,\kappa )$.