'Étale rigidity for motivic spectra' is the statement that (for certain
base schemes S) any p-complete A^1-invariant étale hypersheaf of spectra
on Sm_S is ‚small‘, i.e. comes from the small étale site S_et. This is a deep result proven by Bachmann, building on work of Suslin-Voevodsky, Ayoub, and Cisinski-Déglise. In this talk, I will explain how to generalize this rigidity result to the unstable setting: I will show that certain p-complete A^1-invariant
étale hypersheaves of anima are in fact coming from the small étale
oo-topos. If time permits, I will use this rigidity result to prove an étale
version of Morel’s theorem that strongly A^1-invariant Nisnevich sheaves
of abelian groups are strictly A^1-invariant.