Is there a quasi-isomorphism between a complex of sheaves and its Godement resolutions? I have a doubt, I read somewhere that the Godement resolution of a sheaf $\mathcal{F}$ is a quasi-isomorphism
$\mathcal{F} \rightarrow C^\bullet(\mathcal{F})$.
Just right off the bat when I read that I was like, aren't quasi-isomorphisms supposed to be between complexes? How do I interpret that? Should I interpret it as a quasi-isomorphism
$\mathcal{F}^\bullet \rightarrow C^p(\mathcal{F}^\bullet)$
for all $p$?
 A: Nope. It means that you must interpret the sheaf $\cal{F}$ as a complex concentrated in zero degree (if $\cal{F}$ is a single sheaf). Maybe a "picture" could help:
$$
\begin{array}
{}\vdots   &               & \vdots  \\
\uparrow  &             &  \uparrow \\
    0      &  \rightarrow &  C^2\cal{F}  \\
\uparrow &              &  \uparrow  \\
    0     &  \rightarrow & C^1\cal{F}  \\
\uparrow &              & \uparrow    \\
\cal{F}  &  \rightarrow & C^0\cal{F}
\end{array}
$$
But, if $\cal{F}^\bullet$ is also a complex of sheaves, then $C^\bullet (\cal{F}^\bullet)$ is a double complex. And the quasi-isomorphism is between the complex $\cal{F}^\bullet$ and the total complex $\mathrm{Tot}\ C^\bullet (\cal{F}^\bullet)$ of that double one; that is the (simple) complex which its $n$ degree is
$$
\bigoplus_{p+q=n} C^p\cal{F}^q  \ .
$$
The corresponding "picture":
$$
\begin{array}
{}\vdots   &               & \vdots  &    & \vdots \\
\uparrow  &             &  \uparrow   & &  \uparrow \\
\cal{F}^2   &  \rightarrow &  C^0\cal{F}^2   & \rightarrow & C^1\cal{F}^2 & \rightarrow & \dots  \\
\uparrow &              &  \uparrow    &   &  \uparrow \\
\cal{F}^1       &  \rightarrow & C^0\cal{F}^1 & \rightarrow & C^1\cal{F}^1  &\rightarrow & \dots  \\
\uparrow &              & \uparrow    &   &   \uparrow \\
\cal{F}^0  &  \rightarrow & C^0\cal{F}^0 & \rightarrow & C^1\cal{F}^0  &  \rightarrow & \dots
\end{array}
$$
This is true at least when the complex of sheaves $\cal{F}^\bullet$ is concentrated in positive degrees. Otherwise, you would need some finite cohomological dimension hypothesis on your topological space (or Grothendieck site) where your sheaf is defined.
