The Stacks project

Lemma 14.18.4. Let $U$ be a simplicial set. Let $n \geq 0$ be an integer. The rule

\[ U'_ m = \bigcup \nolimits _{\varphi : [m] \to [i], \ i\leq n} \mathop{\mathrm{Im}}(U(\varphi )) \]

defines a sub simplicial set $U' \subset U$ with $U'_ i = U_ i$ for $i \leq n$. Moreover, all $m$-simplices of $U'$ are degenerate for all $m > n$.

Proof. If $x \in U_ m$ and $x = U(\varphi )(y)$ for some $y \in U_ i$, $i \leq n$ and some $\varphi : [m] \to [i]$ then any image $U(\psi )(x)$ for any $\psi : [m'] \to [m]$ is equal to $U(\varphi \circ \psi )(y)$ and $\varphi \circ \psi : [m'] \to [i]$. Hence $U'$ is a simplicial set. By construction all simplices in dimension $n + 1$ and higher are degenerate. $\square$

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