Lemma 83.8.1. In Situation 83.3.3 and with notation as above there is a complex

$\ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}$

of abelian sheaves on $\mathcal{C}_{total}$ which forms a resolution of the constant sheaf with value $\mathbf{Z}$ on $\mathcal{C}_{total}$.

Proof. We will use the description of the functors $g_{n!}$ in Lemma 83.3.5 without further mention. As maps of the complex we take $\sum (-1)^ i d^ n_ i$ where $d^ n_ i : g_{n!}\mathbf{Z} \to g_{n - 1!}\mathbf{Z}$ is the adjoint to the map $\mathbf{Z} \to \bigoplus _{[n - 1] \to [n]} \mathbf{Z} = g_ n^{-1}g_{n - 1!}\mathbf{Z}$ corresponding to the factor labeled with $\delta ^ n_ i : [n - 1] \to [n]$. Then $g_ m^{-1}$ applied to the complex gives the complex

$\ldots \to \bigoplus \nolimits _{\alpha \in \mathop{Mor}\nolimits _\Delta (, [m])]} \mathbf{Z} \to \bigoplus \nolimits _{\alpha \in \mathop{Mor}\nolimits _\Delta (, [m])]} \mathbf{Z} \to \bigoplus \nolimits _{\alpha \in \mathop{Mor}\nolimits _\Delta (, [m])]} \mathbf{Z}$

on $\mathcal{C}_ m$. In other words, this is the complex associated to the free abelian sheaf on the simplicial set $\Delta [m]$, see Simplicial, Example 14.11.2. Since $\Delta [m]$ is homotopy equivalent to $\Delta $, see Simplicial, Example 14.26.7, and since “taking free abelian sheaf on” is a functor, we see that the complex above is homotopy equivalent to the free abelian sheaf on $\Delta $ (Simplicial, Remark 14.26.4 and Lemma 14.27.2). This complex is acyclic in positive degrees and equal to $\mathbf{Z}$ in degree $0$. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).