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It is known that for a sheaf $\mathcal{F}$ on $X$, we can associate $X_\mathcal{F}$, the étalé space of $\mathcal{F}$ over $X$ such that section of $X_\mathcal{F}$ coincides with section of $\mathcal{F}$.

Is it possible to generalize this concept to derived category of sheaves over $X$ or perverse sheaves over $X$?

There are similar questions asked before, see links below. Yet they focus on abstract nonsense argument. I would like to see concrete constructions specifically for derived category or perverse sheaves.

Etalé space construction for presheaves on a Grothendieck site

How to construct the espace étalé (space of sections) for an arbitrary category?

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  • $\begingroup$ Are you aware of the Grothendieck construction (ncatlab.org/nlab/show/Grothendieck+construction) ? You can do that for presheaves with values in $\infty$-groupoids. That somehow generalises the ordinary construction by putting automorphisms there. I'm however not aware of any construction that somehow would add a topology in that to get some kind of topological $\infty$-groupoid. $\endgroup$
    – user40276
    Dec 12, 2018 at 17:24

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