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
K is split (Simplicial, Definition 14.18.1),
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
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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