Lemma 14.19.12 (018C)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 14: Simplicial Methods
Section 14.19: Coskeleton functors
Lemma 14.19.12 (cite)

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Lemma 14.19.12. Let $U$ , $V$ be $n$ -truncated simplicial objects of a category $\mathcal{C}$ . Then

$\text{cosk}_ n (U \times V) = \text{cosk}_ nU \times \text{cosk}_ nV$

whenever the left and right hand sides exist.

Proof. Let $W$ be a simplicial object. We have

$\begin{eqnarray*} \mathop{\mathrm{Mor}}\nolimits (W, \text{cosk}_ n (U \times V)) & = & \mathop{\mathrm{Mor}}\nolimits (\text{sk}_ n W, U \times V) \\ & = & \mathop{\mathrm{Mor}}\nolimits (\text{sk}_ n W, U) \times \mathop{\mathrm{Mor}}\nolimits (\text{sk}_ nW, V) \\ & = & \mathop{\mathrm{Mor}}\nolimits (W, \text{cosk}_ n U) \times \mathop{\mathrm{Mor}}\nolimits (W, \text{cosk}_ n V) \\ & = & \mathop{\mathrm{Mor}}\nolimits (W, \text{cosk}_ n U \times \text{cosk}_ n V) \end{eqnarray*}$

The lemma follows. $\square$

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