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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