The perverse t-structure for the derived category of complexes of sheaves is certainly a mighty tool for studying cohomology. My question is: does there exist any homotopy-theoretic analogue for it (probably, on some stable homotopy category of sheaves of spectra)?

Higher Algebra, but I don't know how you would define the "support conditions" for sheaves of spectra – Harry Gindi Mar 1 '12 at 11:45