Lemma 25.9.2. Let \mathcal{C} be a site with fibre products. Let X be an object of \mathcal{C}. Let K, L be hypercoverings of X. Let a, b : K \to L be morphisms of hypercoverings. There exists a morphism of hypercoverings c : K' \to K such that a \circ c is homotopic to b \circ c.
Proof. Consider the following commutative diagram
\xymatrix{ K' \ar@{=}[r]^-{def} \ar[rd]_ c & K \times _{(L \times L)} \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L) \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L) \ar[d] \\ & K \ar[r]^{(a, b)} & L \times L }
By the functorial property of \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L) the composition of the horizontal morphisms corresponds to a morphism K' \times \Delta [1] \to L which defines a homotopy between c \circ a and c \circ b. Thus if we can show that K' is a hypercovering of X, then we obtain the lemma. To see this we will apply Lemma 25.7.1 to the pair of morphisms K \to L \times L and \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L) \to L \times L. Condition (1) of Lemma 25.7.1 is satisfied. Condition (2) of Lemma 25.7.1 is true because \mathop{\mathrm{Hom}}\nolimits (\Delta [1], L)_0 = L_1, and the morphism (d^1_0, d^1_1) : L_1 \to L_0 \times L_0 is a covering of \text{SR}(\mathcal{C}, X) by our assumption that L is a hypercovering. To prove condition (3) of Lemma 25.7.1 we use Lemma 25.9.1 above. According to this lemma the morphism \gamma of condition (3) of Lemma 25.7.1 is the morphism
\mathop{\mathrm{Hom}}\nolimits (\Delta [1] \times \Delta [n + 1], L)_0 \longrightarrow \mathop{\mathrm{Hom}}\nolimits (U, L)_0
where U \subset \Delta [1] \times \Delta [n + 1]. According to Lemma 25.8.2 this is a covering and hence the claim has been proven. \square
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