The Stacks project

Lemma 25.12.1. Let $\mathcal{C}$ be a site. Let $K$ be an $r$-truncated simplicial object of $\text{SR}(\mathcal{C})$. The following are equivalent

  1. $K$ is split (Simplicial, Definition 14.18.1),

  2. $f_{\varphi , i} : U_{n, i} \to U_{m, \alpha (\varphi )(i)}$ is an isomorphism for $r \geq n \geq 0$, $\varphi : [m] \to [n]$ surjective, $i \in I_ n$, and

  3. $f_{\sigma ^ n_ j, i} : U_{n, i} \to U_{n + 1, \alpha (\sigma ^ n_ j)(i)}$ is an isomorphism for $0 \leq j \leq n < r$, $i \in I_ n$.

The same holds for simplicial objects if in (2) and (3) we set $r = \infty $.

Proof. The splitting of a simplicial set is unique and is given by the nondegenerate indices $N(I_ n)$ in each degree $n$, see Simplicial, Lemma 14.18.2. The coproduct of two objects $\{ U_ i\} _{i \in I}$ and $\{ U_ j\} _{j \in J}$ of $\text{SR}(\mathcal{C})$ is given by $\{ U_ l\} _{l \in I \amalg J}$ with obvious notation. Hence a splitting of $K$ must be given by $N(K_ n) = \{ U_ i\} _{i \in N(I_ n)}$. The equivalence of (1) and (2) now follows by unwinding the definitions. The equivalence of (2) and (3) follows from the fact that any surjection $\varphi : [m] \to [n]$ is a composition of morphisms $\sigma ^ k_ j$ with $k = n, n + 1, \ldots , m - 1$. $\square$

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