The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's <a href="http://www.math.u-psud.fr/~illusie/vanishing1b.pdf" title="Illusie's article">an issue</a>, Morava <a href="http://arxiv.org/abs/math/0509001" title="Morava's article">mentioned</a> a connection with Bousfield localization. I find the Morava's remarks un-understandable - would someone explain? Exist analoga of the vanishing cycle formalism in "derived", "homotopical" etc. algebraic geometry?