For every complex $K$ in the category of complexes, I need to construct a decreasing filtration of $K$ by subcomplexes $\cdots \supset F^i \supset F^{i+1} \supset \cdots$ such that every factor complex $G^i = F^i/F^{i+1}$ has at most one nonzero cohomology module, which is situated at degree $i$ and is canonically isomorphic to (a) $K^i$, and (b) $H^i(K)$
I thought the canonical truncation/filtration should work for (b) at least, but I'm unable to work the cohomology out of the factor complex in that case. Any help for (a) and (b) will be greatly appreciated
You can indeed consider the usual truncation, that is, let $F_pK$ be the complex that has $\ker \partial$ in degree $p$, nothing in degrees $>p$, and $K^i$ in degrees $<p$. Then the quotient $F_p/F_{p-1}$ looks like
$$0 \longrightarrow K_{p-1}/ \ker (\partial_{p-1}) \longrightarrow \ker \partial_p \longrightarrow 0$$
and has indeed homology concentrated in degree $p$ where it is $H^p(K)$. If you instead put $K^p$ in degree $p$, you will get just
$$0 \longrightarrow 0 \longrightarrow K^p \longrightarrow 0.$$