Let $\mathcal A$ be an abelian category. Is there any useful criteria how to identify when a cochain complex is quasi-isomorphic to a complex concentrated in a single degree, that is the two are isomorphic in the derived category $D(\mathcal A)$? Such criteria could be about employing homological dimension or a certain class of cochain complexes of which such identification is tautological for the class. I don't mind criteria identifying when they are homotopic too.
My apology if similar question had been asked before as I couldn't find it to the best of my effort.
Suppose that $X$ has cohomology concentrated in a single degree. Without loss of generality let that degree be $0$. Consider the complex $$\tau_0X:\cdots \to X_2\to X_1 \to Z_0X \to 0 $$
This complex is a subcomplex of $X$, and the inclusion $\tau_0X \to X$ is a quasi isomorphism.
Now consider the complex
$$H_0 :\cdots \to 0 \to 0 \to H_0(X) \to 0 $$
There is a projection $\tau_0X \to H_0 $, which is a quasi-isomorphism. In conclusion, $X$ is weakly equivalent to $H_0$.
Add. The complex $\tau_j X$ with $(\tau_j X)_i = X_i$ if $i>j$, $(\tau_j X)_j = Z_jX$ and $(\tau_j X)_i = 0$ if $j<i$ is called the (good) truncation of $X$. Note that it is a subcomplex of $X$ with the property that the inclusion $\iota_j : \tau_j X\longrightarrow X$ induces an isomorphism in homology degrees $i\geqslant j$.