Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective over an affine scheme and $M$ is an étale sheaf, we know that the Cech cohomology $\check{\mathrm{H}}^\bullet(X,M)$ coincides with the sheaf cohomology $H^\bullet(X,M)$.

I wonder what happens when $M$ is a (possibly unbounded) complex. Is there a Cech-like way of describing the (hyper)cohomology $H^\bullet(X,M^\bullet)$ or, even better, the complex $\mathsf{R}f_* M^\bullet$ for some map $f$?

If that's necessary, an answer using hypercovers (which I know very little about) would also be interesting!

Let $X$ be a topological space (or a site) and let $M$ be a sheaf on $X$. If $X$ is paracompact, or if $X$ is a noetherian separated scheme and $M$ is quasi-coherent, or if $X$ is quasi-projective over an affine scheme and $M$ is an étale sheaf, we know that the Čech cohomology $\smash{\check{\mathrm{H}}}^\bullet(X,M)$ coincides with the sheaf cohomology $H^\bullet(X,M)$.

I wonder what happens when $M$ is a (possibly unbounded) complex. Is there a Čech-like way of describing the (hyper)cohomology $H^\bullet(X,M^\bullet)$ or, even better, the complex $\mathsf{R}f_* M^\bullet$ for some map $f$?

If that's necessary, an answer using hypercovers (which I know very little about) would also be interesting!