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What is the proper terminology for a complex of sheaves $\mathcal F^\bullet$ whose homology sheaves $\mathcal H^i\mathcal F^\bullet$ vanish for $i\ne 0$?

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One can call it pure object.

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In the context of t-structures, it's not unreasonable to call this just a "sheaf". That is, in the natural t-structure on the derived category, the core is exactly the full subcategory of these things. In the perverse t-structure it is, by definition, the category of perverse sheaves. If one can say that a complex with vanishing positive and negative perverse cohomology is a "perverse sheaf" surely one can also say that a complex satisfying the vanishing of positive and negative cohomology is just a sheaf.

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