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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)?

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Lurie gives a definition of a t-structure for stable $\infty$-categories in 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
    
So, Lurie didn't define anything like this? –  Mikhail Bondarko Mar 1 '12 at 12:25
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I think that the question is not whether $t$-structures can be defined for enhancements of triangulated categories (such as $\infty$-categories), but whether perverse $t$-structures can be defined for ordinary triangulated categories of purely topological origin. Mikhail, am I right? –  Fernando Muro Mar 1 '12 at 13:50
    
Yes; I am not very much interested in enhancements of any sort. Yet I am not sure that one could call (some) triangulated category of etale sheaves of spectra a category of purely topological origin.:) –  Mikhail Bondarko Mar 1 '12 at 14:40
    
I meant it cannot be embedded in the homotopy category of an additive category (the existence of such an embedding is often taken as a definition of algebraic triangulated category). –  Fernando Muro Mar 1 '12 at 15:24

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