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It is known that, given an abelian category $\mathcal B$, the derived category $\mathrm D (\mathcal B)$ is a triangulated category. In particular, given a distinguished triangle $E\to F\to G \to E[1]$ in $\mathrm D ( \mathcal B)$, taking cohomology of complexes gives a long exact sequence $$ \cdots \to H^*(E) \to H^*(F) \to H^*(G) \to H^{*+1}(E) \to \cdots $$

For a general triangulated category, there is no notion of "complexes". Nevertheless, we have something similar provided a bounded t-structure. (Reference: Dirichlet Branes and Mirror Symmetry section 4.4)

Indeed, suppose $\mathcal A$ is the heart of a bounded t-structure on a triangulated category $\mathrm D$. Then it is known that for every nonzero object $E\in \mathrm D$ we can define the so-called cohomology objects $H^i(E)\in \mathcal A$ of $E$ in the given t-structure.

Question:

(1) If $E\to F\to G \to E[1]$ is a distinguished triangle in a triangulated category $\mathrm D$, then can we still have a long exact sequence as above? Can we find a cohomological functor $\mathrm D \to \mathcal A$ to explain this?

(2) What is the motivation of the definition of cohomology objects $H^i(E)$ of $E$ given a t-structure? (Especially, do you know how to establish the Lemma 4.56 in the reference mentioned above?) Thanks

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(1) The functor $H^0$ is cohomological [BBD, Théorème 1.3.6].

(2) A t-structure is, basically, a way to determine an abelian subcategory inside a triangulated category together with a cohomological functor with values in this subcategory. This permits a formulation within the derived category of sheaves of perverse sheaves.

Perverse sheaves are complexes associated to a stratification and are associated to the theory of Intersection Homology of Goresky and McPherson. This theory was developed to recover duality on stratified spaces.

Moreover, this general formalism has allowed Beĭlinson, Bernstein, Deligne and Gabber to transport the construction of intersection cohomology to the context of étale cohomology of algebraic varieties (possibly with singularities).

[BBD] Beĭlinson, A. A.; Bernstein, J.; Deligne, P.: Faisceaux pervers. Analysis and topology on singular spaces, I (Luminy, 1981), 5–171, Astérisque, 100, Soc. Math. France, Paris, 1982.

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