Algebra and Geometry Seminar

Étale homotopy theory was invented by Artin and Mazur in the 1960s as a way to associate to a scheme S, a homotopy type with fundamental group the étale fundamental group of S and whose cohomology captures the étale cohomology of S with locally constant constructible coefficients. In this talk we'll explain how to construct a stratified refinement of the étale homotopy type that classifies constructible étale sheaves of spaces. We'll also explain how this refinement gives rise to a new, concrete definition of the étale homotopy type. We'll then explain how to use condensed math to upgrade this result from discrete rings to rings with a topology such as Z_\ell, Q_\ell, or F_q[[t]]. This is joint work with Clark Barwick and Saul Glasman.