Let X be a locally topologically noetherian scheme. In their paper on the proétale topology, Bhatt and Scholze defined the proétale fundamental group π1proét(X). The profinite completion of π1proét(X) recovers the usual étale fundamental group. Moreover, π1proét(X) agrees with π1ét(X) when X is normal, but π1proét(X) has the better property that it classifies Qp-local systems. In this talk, we’ll explain how to use condensed mathematics to define a “condensed homotopy type” whose fundamental group refines the proétale fundamental group. We’ll also talk about two related “condensed homotopy types”: one for adic spaces and one in the setting of logic. All three are special cases of a more general invariant of ∞-topoi. For adic spaces, the fundamental group refines the de Jong fundamental group, and in logic, the fundamental group recovers the Lascar group of a complete first-order theory.
These results come from several projects, carried out jointly with Holzschuh–Lara–Mair–Martini–Wolf, Achinger–Holzschuh–Lara–Mair–Wolf, and Damaj–Zhang.