If $\mathcal{H}$ is the heart of a bounded t-structure in a triangulated category $\mathcal{T}$, then for every object $E$ in $\mathcal{T}$ there exists a finite sequence of integers $k\_1>k\_2>\dots >k\_n$ and a collection of triangles in $\mathcal{T}$
$$
E\_{i-1}\stackrel{\phi\_i}{\to}E\_i\to
A\_i \to E\_{i-1}[1]
$$
for $i=0,\dots,n$,
with $E\_0=0$, $E\_n=E$ and $A\_i\in \mathcal{H}[k\_i]$ for any $i=1,\dots,n$. This can be seen as a Postnikov tower for the initial morphism $0 \to E$, so by analogy with what happens in algebraic topology one is led to wonder whether more generally one has Postnikov towers for arbitrary morphisms in a bounded t-structure, i.e., the same as above, but this time with $E\_0=X$, $E\_n=Y$ and $\phi\_n\circ\phi\_{n-1}\circ\cdots\circ \phi\_1$ equal to some given morphism $f:X\to Y$.


I have so far been unable to rigorously prove (or disprove) such a statement nor to locate it in the literature.
As usual any suggestion in either direction will be appreciated.



Postnikov towers in bounded t-structures
Let $F$ be the fiber of $f:X \to Y$ (i.e. the cone shifted by $[-1]$) so that we have a distinguished triangle
$$
F \to X \to Y.
$$
Let $0 = F\_0 \to F\_1 \to \dots \to F\_n = F$ be its Postnikov tower. Let $E\_i$ be the cone of the map $F\_i \to X$ (the composition of $F\_i \to F$ and $F \to X$). Then $E\_0 = Cone(0 \to X) = X$ and $E\_n = Cone(F \to X) \cong Y$. On the other hand, by octahedron axiom 
$$
Cone(E\_{i-1} \to E\_i) = Cone (F\_{i-1} \to F\_i)[1],
$$
so I think this is what you want.


