Quasifibrations in étale homotopy theory

Let S be a noetherian and normal scheme and \(f\colon X \to S\) a geometric fibration (e.g. a smooth and proper morphism) with geometrically connected fibres.

In [Fri73], Friedlander proved that, after completion away from char(S), the étale homotopy type of a geometric fibre of f coincides with the homotopy fibre of the induced map on étale homotopy types.

In this talk, I will explain a more conceptual proof (strategy) of Friedlanders result that works for arbitrary qcqs schemes S. This is joint work in progress with Alexander Schmidt and Jakob Stix.