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!



Is there a Čech-like way of computing $H^\bullet(X,M^\bullet)$ or even $\mathsf{R}f_* M^\bullet$?

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> Is there a Cech-like way of describing the (hyper)cohomology H∙(X,M∙) or, even better, the complex Rf∗M∙ for some map f?
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> 
> 


Yes, the [Verdier hypercovering theorem](https://ncatlab.org/nlab/show/hypercover#DescentAndCohomology) allows one to compute sheaf cohomology on any site in terms of hypercovers.


Specifically, suppose $M$ is a presheaf of unbounded cochain complexes on a site $S$ and $X∈S$.


Denote by $H/X$ the category whose objects are hypercovers of $X$ and morphisms are commutative triangles.


Denote by $\def\Ch{{\sf Ch}}\def\op{{\sf op}}H/X^\op→\Ch$ the (contravariant) functor that sends a hypercover $U→X$ to the mapping chain complex $\def\Map{\mathop{\rm Map}}\Map(U,M)$
and on morphisms is given by the restriction maps.


Denote by $C(X,M)$ the colimit of the functor $H/X^\op→\Ch$.
Since hypercovers can be pulled back, a morphism $X→Y$ induces a map $C(Y,M)→C(X,M)$, which turns $C(-,M)$ into a presheaf of chain complexes.


The canonical map $M→C(-,M)$ is a local quasi-isomorphism
and its target satisfies the homotopy coherent descent property with respect to all hypercovers.
Therefore, the cohomology of $C(-,M)$ computes the sheaf cohomology with coefficients in $M$.


