The Stacks project

Remark 25.12.9. Let $\mathcal{C}$ be a site. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Assume $\mathcal{C}$ has fibre products and let $K$ be a hypercovering of $X$. Write $K_ n = \{ U_{n, i}\} _{i \in I_ n}$. Suppose that

  1. $U_ n = \coprod _{i \in I_ n} U_{n, i}$ exists,

  2. given morphisms $(\alpha , f_ i) : \{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ and $(\beta , g_ k) : \{ W_ k\} _{k \in K} \to \{ V_ j\} _{j \in J}$ in $\text{SR}(\mathcal{C})$ such that $U = \coprod U_ i$, $V = \coprod V_ j$, and $W = \coprod W_ j$ exist, then $U \times _ V W = \coprod _{(i, j, k), \alpha (i) = j = \beta (k)} U_ i \times _{V_ j} W_ k$,

  3. if $(\alpha , f_ i) : \{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ is a covering in the sense of Definition 25.3.1 and $U = \coprod U_ i$ and $V = \coprod V_ j$ exist, then the corresponding morphism $U \to V$ of Remark 25.12.7 is a covering of $\mathcal{C}$.

Then we get another simplicial object $L$ of $\text{SR}(\mathcal{C})$ with $L_ n = \{ U_ n\} $, see Remark 25.12.7. Now we claim that $L$ is a hypercovering of $X$. To see this we check conditions (1), (2) of Definition 25.3.3. Condition (1) follows from (c) and (1) for $K$ because (1) for $K$ says $K_0 = \{ U_{0, i}\} _{i \in I_0}$ is a covering of $\{ X\} $ in the sense of Definition 25.3.1. Condition (2) follows because $\mathcal{C}/X$ has all finite limits hence $\text{SR}(\mathcal{C}/X)$ has all finite limits, and condition (b) says the construction of “taking disjoint unions” commutes with these fimite limits. Thus the morphism

\[ L_{n + 1} \longrightarrow (\text{cosk}_ n \text{sk}_ n L)_{n + 1} \]

is a covering as it is the consequence of applying our “taking disjoint unions” functor to the morphism

\[ K_{n + 1} \longrightarrow (\text{cosk}_ n \text{sk}_ n K)_{n + 1} \]

which is assumed to be a covering in the sense of Definition 25.3.1 by (2) for $K$. This makes sense because property (b) in particular assures us that if we start with a finite diagram of semi-representable objects over $X$ for which we can take disjoint unions, then the limit of the diagram in $\text{SR}(\mathcal{C}/X)$ still is a semi-representable object over $X$ for which we can take disjoint unions.

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