groups on $Y$ will be denoted by ${\mathcal{F}}$ and not by ${\mathcal{F}}^{{\bullet}}$; likewise, the cohomology groups of $Y$ with values in ${\mathcal{F}}$ will be denoted by $H^k(Y,{\mathcal{F}})$.

Assume that ${\varepsilon}:X_{{\bullet}}\rightarrow Y$ is an augmented $n$-semisimplicial space and ${\mathcal{F}}^{{\bullet}}$ is a sheaf of abelian groups on $X_{{\bullet}}$. The sheaves ${\varepsilon}_*{\mathcal{C}}_{Gdm}^p({\mathcal{F}}^q)$ form a double complex of sheaves of abelian groups on $Y$, whose associated simple complex gives

$$R{\varepsilon}_*{\mathcal{F}}^{{\bullet}}:=s[{\varepsilon}_*{\mathcal{C}}_{Gdm}^{{\bullet}}({\mathcal{F}}^{{\bullet}})].$$

One can prove that the hypercohomology of the latter complex coincides with the cohomology of $X_{{\bullet}}$ with values in ${\mathcal{F}}^{{\bullet}}$, i.e.

$$\label{eq:isocohom}
\mathbb{H}^k(Y,R{\varepsilon}_*{\mathcal{F}}^{{\bullet}})=H^k(X_{{\bullet}},{\mathcal{F}}^{{\bullet}})\hspace{0.5cm}\text{for any }k.$$

[@De Definition 5.3.2] An augmented $n$-semisimplicial space ${\varepsilon}:X_{{\bullet}}\rightarrow Y$ is *of cohomological descent* if for any sheaf of abelian groups ${\mathcal{F}}$ on $Y$ the natural adjunction morphism

$$\label{eq:CohomDesc}
{\mathcal{F}}\rightarrow R{\varepsilon}_*{\varepsilon}^{-1}{\mathcal{F}}$$

is an isomorphism in $D_+(Sh(Y))$.

1.  Observe that if ${\varepsilon}:X_{{\bullet}}\rightarrow Y$ is of cohomological descent then $H^k(Y,{\mathcal{F}})\simeq H^k(X_{{\bullet}},{\varepsilon}^{-1}{\mathcal{F}})$ for any $k$ and for any sheaf of abelian groups ${\mathcal{F}}$ on $Y$.

2.  If $X_{{\square}}$ is an $(n+1)$-cubical space and ${\mathcal{F}}^{{\square}}$ is a sheaf of abelian groups on $X$, by point 3. of Remark \[rmk:cubobj\] to this data we can associate an augmented $n$-semisimplicial space ${\varepsilon}:X_{{\bullet}}\rightarrow X_{\emptyset}$ and a sheaf of abelian groups ${\mathcal{F}}^{{\bullet}}$ on it. We set

    $$\label{eq:cono}
    C^{{\bullet}}(X_{{\square}},{\mathcal{F}}^{{\square}}):=\text{Cone}^{{\bullet}}[{\mathcal{F}}^{\emptyset}\rightarrow 