Lemma 25.7.1. Let \mathcal{C} be a site with fibre products. Let X be an object of \mathcal{C}. Let K, L, M be simplicial objects of \text{SR}(\mathcal{C}, X). Let a : K \to L, b : M \to L be morphisms. Assume
K is a hypercovering of X,
the morphism M_0 \to L_0 is a covering, and
for all n \geq 0 in the diagram
\xymatrix{ M_{n + 1} \ar[dd] \ar[rr] \ar[rd]^\gamma & & (\text{cosk}_ n \text{sk}_ n M)_{n + 1} \ar[dd] \\ & L_{n + 1} \times _{(\text{cosk}_ n \text{sk}_ n L)_{n + 1}} (\text{cosk}_ n \text{sk}_ n M)_{n + 1} \ar[ld] \ar[ru] & \\ L_{n + 1} \ar[rr] & & (\text{cosk}_ n \text{sk}_ n L)_{n + 1} }
the arrow \gamma is a covering.
Then the fibre product K \times _ L M is a hypercovering of X.
Proof.
The morphism (K \times _ L M)_0 = K_0 \times _{L_0} M_0 \to K_0 is a base change of a covering by (2), hence a covering, see Lemma 25.3.2. And K_0 \to \{ X \to X\} is a covering by (1). Thus (K \times _ L M)_0 \to \{ X \to X\} is a covering by Lemma 25.3.2. Hence K \times _ L M satisfies the first condition of Definition 25.3.3.
We still have to check that
K_{n + 1} \times _{L_{n + 1}} M_{n + 1} = (K \times _ L M)_{n + 1} \longrightarrow (\text{cosk}_ n \text{sk}_ n (K \times _ L M))_{n + 1}
is a covering for all n \geq 0. We abbreviate as follows: A = (\text{cosk}_ n \text{sk}_ n K)_{n + 1}, B = (\text{cosk}_ n \text{sk}_ n L)_{n + 1}, and C = (\text{cosk}_ n \text{sk}_ n M)_{n + 1}. The functor \text{cosk}_ n \text{sk}_ n commutes with fibre products, see Simplicial, Lemma 14.19.13. Thus the right hand side above is equal to A \times _ B C. Consider the following commutative diagram
\xymatrix{ K_{n + 1} \times _{L_{n + 1}} M_{n + 1} \ar[r] \ar[d] & M_{n + 1} \ar[d] \ar[rd]_\gamma \ar[rrd] & & \\ K_{n + 1} \ar[r] \ar[rd] & L_{n + 1} \ar[rrd] & L_{n + 1} \times _ B C \ar[l] \ar[r] & C \ar[d] \\ & A \ar[rr] & & B }
This diagram shows that
K_{n + 1} \times _{L_{n + 1}} M_{n + 1} = (K_{n + 1} \times _ B C) \times _{(L_{n + 1} \times _ B C), \gamma } M_{n + 1}
Now, K_{n + 1} \times _ B C \to A \times _ B C is a base change of the covering K_{n + 1} \to A via the morphism A \times _ B C \to A, hence is a covering. By assumption (3) the morphism \gamma is a covering. Hence the morphism
(K_{n + 1} \times _ B C) \times _{(L_{n + 1} \times _ B C), \gamma } M_{n + 1} \longrightarrow K_{n + 1} \times _ B C
is a covering as a base change of a covering. The lemma follows as a composition of coverings is a covering.
\square
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