have a morphism $d_{JI}:X_I\rightarrow X_J$. We can now define the morphism

    $$d_{{\beta}}:X_r\rightarrow X_s\hspace{0.5cm}\text{s.t. }(d_{{\beta}})_{|X_I}=d_{{\beta}(I)I}.$$

    Since for any $I\in{\square}_{n+1}$ we have a morphism $d_{\emptyset I}:X_I\rightarrow X_{\emptyset}$ we obtain the desired augmentation by setting ${\varepsilon}_{|X_I}:=d_{\emptyset I}$.

The category $TopAbSh$ has objects the pairs $(X,{\mathcal{F}})$ where $X$ is a topological space and ${\mathcal{F}}$ is a sheaf of abelian groups on $X$, and as morphisms the pairs $(f,f^{\#}):(X,{\mathcal{F}})\rightarrow (Y,{\mathcal{G}})$ where $f:X\rightarrow Y$ is a continuous function and $f^{\#}:{\mathcal{G}}\rightarrow f_*{\mathcal{F}}$ is a morphism of sheaves of abelian groups on $Y$. A *sheaf of abelian groups* ${\mathcal{F}}^{{\bullet}}$ (resp. ${\mathcal{F}}^{{\square}}$) on an $n$-semisimplicial space (resp. on an $n$-cubical space) is just an $n$-semisimplicial (resp. $n$-cubical) $TopAbSh$-object. In a similar manner, we can define complexes and resolutions of sheaves of abelian groups on $n$-semisimplicial or $n$-cubical spaces.

Given a sheaf of abelian groups ${\mathcal{F}}^{{\bullet}}$ on an $n$-semisimplicial space $X_{{\bullet}}$, it is possible to define the cohomology of $X_{{\bullet}}$ with values in ${\mathcal{F}}^{{\bullet}}$; indeed, using the Godement resolutions of each ${\mathcal{F}}^m$ and differentials coming from the face maps of ${\triangle}_n$, one obtains a double complex $F^{{\bullet},{\bullet}}$ and sets

$$H^k(X_{{\bullet}},{\mathcal{F}}^{{\bullet}}):=H^k(s(F^{{\bullet},{\bullet}}))$$

where $s(F^{{\bullet},{\bullet}})$ is the simple complex associated to $F^{{\bullet},{\bullet}}$.

If $Y$ is a constant $n$-semisimplicial space, any sheaf of abelian 