# the cohomology objects in the t-structure and long exact sequence

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

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