Lemma 21.9.4. Let \mathcal{C} be a category. Let \mathcal{U} = \{ f_ i : U_ i \to U\} _{i \in I} be a family of morphisms with fixed target such that all fibre products U_{i_0} \times _ U \ldots \times _ U U_{i_ p} exist in \mathcal{C}. The chain complex \mathbf{Z}_{\mathcal{U}, \bullet } of presheaves of Lemma 21.9.3 above is exact in positive degrees, i.e., the homology presheaves H_ i(\mathbf{Z}_{\mathcal{U}, \bullet }) are zero for i > 0.
Proof. Let V be an object of \mathcal{C}. We have to show that the chain complex of abelian groups \mathbf{Z}_{\mathcal{U}, \bullet }(V) is exact in degrees > 0. This is the complex
\xymatrix{ \ldots \ar[d] \\ \bigoplus _{i_0i_1i_2} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U_{i_0} \times _ U U_{i_1} \times _ U U_{i_2}) ] \ar[d] \\ \bigoplus _{i_0i_1} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U_{i_0} \times _ U U_{i_1}) ] \ar[d] \\ \bigoplus _{i_0} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(V, U_{i_0}) ] \ar[d] \\ 0 }
For any morphism \varphi : V \to U denote \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_ i) = \{ \varphi _ i : V \to U_ i \mid f_ i \circ \varphi _ i = \varphi \} . We will use a similar notation for \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0} \times _ U \ldots \times _ U U_{i_ p}). Note that composing with the various projection maps between the fibred products U_{i_0} \times _ U \ldots \times _ U U_{i_ p} preserves these morphism sets. Hence we see that the complex above is the same as the complex
\xymatrix{ \ldots \ar[d] \\ \bigoplus _\varphi \bigoplus _{i_0i_1i_2} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0} \times _ U U_{i_1} \times _ U U_{i_2}) ] \ar[d] \\ \bigoplus _\varphi \bigoplus _{i_0i_1} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0} \times _ U U_{i_1}) ] \ar[d] \\ \bigoplus _\varphi \bigoplus _{i_0} \mathbf{Z}[ \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) ] \ar[d] \\ 0 }
Next, we make the remark that we have
\mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0} \times _ U \ldots \times _ U U_{i_ p}) = \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) \times \ldots \times \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_ p})
Using this and the fact that \mathbf{Z}[A] \oplus \mathbf{Z}[B] = \mathbf{Z}[A \amalg B] we see that the complex becomes
\xymatrix{ \ldots \ar[d] \\ \bigoplus _\varphi \mathbf{Z}\left[ \coprod _{i_0i_1i_2} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) \times \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_1}) \times \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_2}) \right] \ar[d] \\ \bigoplus _\varphi \mathbf{Z}\left[ \coprod _{i_0i_1} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) \times \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_1}) \right] \ar[d] \\ \bigoplus _\varphi \mathbf{Z}\left[ \coprod _{i_0} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_{i_0}) \right] \ar[d] \\ 0 }
Finally, on setting S_\varphi = \coprod _{i \in I} \mathop{\mathrm{Mor}}\nolimits _\varphi (V, U_ i) we see that we get
\bigoplus \nolimits _\varphi \left(\ldots \to \mathbf{Z}[S_\varphi \times S_\varphi \times S_\varphi ] \to \mathbf{Z}[S_\varphi \times S_\varphi ] \to \mathbf{Z}[S_\varphi ] \to 0 \to \ldots \right)
Thus we have simplified our task. Namely, it suffices to show that for any nonempty set S the (extended) complex of free abelian groups
\ldots \to \mathbf{Z}[S \times S \times S] \to \mathbf{Z}[S \times S] \to \mathbf{Z}[S] \xrightarrow {\Sigma } \mathbf{Z} \to 0 \to \ldots
is exact in all degrees. To see this fix an element s \in S, and use the homotopy
n_{(s_0, \ldots , s_ p)} \longmapsto n_{(s, s_0, \ldots , s_ p)}
with obvious notations. \square
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