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The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned 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?

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# higher vanishing cycles

The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned 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?