Condensed Learning Seminar

Introduce the notion of hypersheaf and discuss its basic properties and equivalent characterizations. In particular, discuss the notions of homotopy and cohomological dimensions and their relation to hyperdescent. Sketch the proof of the fact for finite dimensional qcqs schemes and analytic adic space, the corresponding Nisnevich topos has finite homotopy dimension. Explain the counterexample [CM21, Example 4.15] in the étale case. Sketch the proof of the hypercompleteness criterion in terms of topos theoretic points (see [CM21, Theorem 4.36] and its adic analog [And23, Satz B.31]) and use it to prove that for finite dimensional analytic adic spaces, localizing invariants satisfy étale hyperdescent after chromatic localization (see [And23, Satz 5.14]) using the algebraic analog as input (see [CM21, Theorem 7.14]).