In the proof of SGA 4, XVII, Proposition 4.2.10, the author makes a "flat Cartan-Eilenberg resolution" of a complex of sheaves.
Soit $K$ un complexe vérifiant (a). ... Soit $K_1^*$ une résolution plate de Cartan-Eilenberg et posons $K_2 = \tau_{\ge -N}^{\prime\prime} K_1$ (troncature dans le sens de la nouvelle différentielle). La flèche composée de $K_2$ dans $K$ est un quasi-isomorphisme, ...
But I only find the definition of projective and injective Cartan-Eilenberg resolutions. So what is flat Cartan-Eilenberg resolution?
I found a possible way to work it out (I don't know whether it's correct). For $K^\bullet$ bounded above, just take a flat resolution of $K^\bullet$. For $K^\bullet$ unbounded, we have $N < \infty$. There is a complex of flat sheaves $P_0^\bullet$, which is split exact, with a surjection $\varepsilon: P_0^\bullet \to K^\bullet$. So we have a quasi-isomorphism $K^\bullet \cong (\ker\varepsilon)^\bullet[1]$. Inductively, we can construct a complex (of complexes)
$$\cdots \xrightarrow{d_{n+1}} P_n^\bullet \xrightarrow{d_{n}} \cdots \to P_1^\bullet \xrightarrow{d_1} P_0^\bullet \xrightarrow{d_0=\varepsilon} K^\bullet \to 0 \to \cdots$$
which is exact and satisfies $(\ker d_n)^\bullet[n+1] \xrightarrow{\mathrm{qis}} K^\bullet$. The double complex given by $P_{\bullet}^\bullet$ satisfies our requirement.