On weakly \'etale morphisms

We show that the weakly \'etale morphisms, used to define the pro-\'etale site of a scheme, are characterized by a lifting property similar to the one which characterizes formally \'etale morphisms. In order to prove this, we prove a theorem called Henselian descent which is a "Henselized version" of the fact that a scheme defines a sheaf for the fpqc topology. Finally, we show that weakly \'etale algebras over regular rings arising in geometry are ind-\'etale and that weakly \'etale algebras do not always lift along surjective ring homomorphisms.