Lemma 14.21.2 (018M)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 14: Simplicial Methods
Section 14.21: Left adjoints to the skeleton functors
Lemma 14.21.2 (cite)

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Lemma 14.21.2. Let $\mathcal{C}$ be a category. Let $U$ be an $m$ -truncated simplicial object of $\mathcal{C}$ . For any $n \leq m$ the colimit

$\mathop{\mathrm{colim}}\nolimits _{([n]/\Delta )_{\leq m}^{opp}} U(n)$

exists and is equal to $U_ n$ .

Proof. This is so because the category $([n]/\Delta )_{\leq m}$ has an initial object, namely $\text{id} : [n] \to [n]$ . $\square$

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