The Stacks project

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24.8 Adding simplices

In this section we prove some technical lemmas which we will need later. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. As we pointed out in Section 24.7 above, the objects $U \times K$ and $\mathop{\mathrm{Hom}}\nolimits (U, K)$ for certain simplicial sets $U$ and any simplicial object $K$ of $\text{SR}(\mathcal{C}, X)$ are defined. See Simplicial, Sections 14.13 and 14.17.

Lemma 24.8.1. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $U \subset V$ be simplicial sets, with $U_ n, V_ n$ finite nonempty for all $n$. Assume that $U$ has finitely many nondegenerate simplices. Suppose $n \geq 0$ and $x \in V_ n$, $x \not\in U_ n$ are such that

  1. $V_ i = U_ i$ for $i < n$,

  2. $V_ n = U_ n \cup \{ x\} $,

  3. any $z \in V_ j$, $z \not\in U_ j$ for $j > n$ is degenerate.

Then the morphism

\[ \mathop{\mathrm{Hom}}\nolimits (V, K)_0 \longrightarrow \mathop{\mathrm{Hom}}\nolimits (U, K)_0 \]

of $\text{SR}(\mathcal{C}, X)$ is a covering.

Proof. If $n = 0$, then it follows easily that $V = U \amalg \Delta [0]$ (see below). In this case $\mathop{\mathrm{Hom}}\nolimits (V, K)_0 = \mathop{\mathrm{Hom}}\nolimits (U, K)_0 \times K_0$. The result, in this case, then follows from Lemma 24.3.2.

Let $a : \Delta [n] \to V$ be the morphism associated to $x$ as in Simplicial, Lemma 14.11.3. Let us write $\partial \Delta [n] = i_{(n-1)!} \text{sk}_{n - 1} \Delta [n]$ for the $(n - 1)$-skeleton of $\Delta [n]$. Let $b : \partial \Delta [n] \to U$ be the restriction of $a$ to the $(n - 1)$ skeleton of $\Delta [n]$. By Simplicial, Lemma 14.21.7 we have $V = U \amalg _{\partial \Delta [n]} \Delta [n]$. By Simplicial, Lemma 14.17.5 we get that

\[ \xymatrix{ \mathop{\mathrm{Hom}}\nolimits (V, K)_0 \ar[r] \ar[d] & \mathop{\mathrm{Hom}}\nolimits (U, K)_0 \ar[d] \\ \mathop{\mathrm{Hom}}\nolimits (\Delta [n], K)_0 \ar[r] & \mathop{\mathrm{Hom}}\nolimits (\partial \Delta [n], K)_0 } \]

is a fibre product square. Thus it suffices to show that the bottom horizontal arrow is a covering. By Simplicial, Lemma 14.21.11 this arrow is identified with

\[ K_ n \to (\text{cosk}_{n - 1} \text{sk}_{n - 1} K)_ n \]

and hence is a covering by definition of a hypercovering. $\square$

Lemma 24.8.2. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Let $U \subset V$ be simplicial sets, with $U_ n, V_ n$ finite nonempty for all $n$. Assume that $U$ and $V$ have finitely many nondegenerate simplices. Then the morphism

\[ \mathop{\mathrm{Hom}}\nolimits (V, K)_0 \longrightarrow \mathop{\mathrm{Hom}}\nolimits (U, K)_0 \]

of $\text{SR}(\mathcal{C}, X)$ is a covering.

Proof. By Lemma 24.8.1 above, it suffices to prove a simple lemma about inclusions of simplicial sets $U \subset V$ as in the lemma. And this is exactly the result of Simplicial, Lemma 14.21.8. $\square$

Lemma 24.8.3. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Then

  1. $K_ n$ is a covering of $X$ for each $n \geq 0$,

  2. $d^ n_ i : K_ n \to K_{n - 1}$ is a covering for all $n \geq 1$ and $0 \leq i \leq n$.

Proof. Recall that $K_0$ is a covering of $X$ by Definition 24.3.3 and that this is equivalent to saying that $K_0 \to \{ X \to X\} $ is a covering in the sense of Definition 24.3.1. Hence (1) follows from (2) because it will prove that the composition $K_ n \to K_{n - 1} \to \ldots \to K_0 \to \{ X \to X\} $ is a covering by Lemma 24.3.2.

Proof of (2). Observe that $\mathop{Mor}\nolimits (\Delta [n], K)_0 = K_ n$ by Simplicial, Lemma 14.17.4. Therefore (2) follows from Lemma 24.8.2 applied to the $n + 1$ different inclusions $\Delta [n - 1] \to \Delta [n]$. $\square$

Remark 24.8.4. A useful special case of Lemmas 24.8.2 and 24.8.3 is the following. Suppose we have a category $\mathcal{C}$ having fibre products. Let $P \subset \text{Arrows}(\mathcal{C})$ be a subset stable under base change, stable under composition, and containing all isomorphisms. Then one says a $P$-hypercovering is an augmentation $a : U \to X$ from a simplicial object of $\mathcal{C}$ such that

  1. $U_0 \to X$ is in $P$,

  2. $U_1 \to U_0 \times _ X U_0$ is in $P$,

  3. $U_{n + 1} \to (\text{cosk}_ n\text{sk}_ n U)_{n + 1}$ is in $P$ for $n \geq 1$.

The category $\mathcal{C}/X$ has all finite limits, hence the coskeleta used in the formulation above exist (see Categories, Lemma 4.18.4). Then we claim that the morphisms $U_ n \to X$ and $d^ n_ i : U_ n \to U_{n - 1}$ are in $P$. This follows from the aforementioned lemmas by turning $\mathcal{C}$ into a site whose coverings are $\{ f : V \to U\} $ with $f \in P$ and taking $K$ given by $K_ n = \{ U_ n \to X\} $.


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