The Stacks project

Lemma 14.21.11. Let $\mathcal{C}$ be a category with finite coproducts and finite limits. Let $V$ be a simplicial object of $\mathcal{C}$. In this case

\[ (\text{cosk}_ n \text{sk}_ n V)_{n + 1} = \mathop{\mathrm{Hom}}\nolimits (i_{n !}\text{sk}_ n \Delta [n + 1], V)_0. \]

Proof. By Lemma 14.13.4 the object on the left represents the functor which assigns to $X$ the first set of the following equalities

\begin{eqnarray*} \mathop{\mathrm{Mor}}\nolimits (X \times \Delta [n + 1], \text{cosk}_ n \text{sk}_ n V) & = & \mathop{\mathrm{Mor}}\nolimits (X \times \text{sk}_ n \Delta [n + 1], \text{sk}_ n V) \\ & = & \mathop{\mathrm{Mor}}\nolimits (X \times i_{n !} \text{sk}_ n \Delta [n + 1], V). \end{eqnarray*}

The object on the right in the formula of the lemma is represented by the functor which assigns to $X$ the last set in the sequence of equalities. This proves the result.

In the sequence of equalities we have used that $\text{sk}_ n (X \times \Delta [n + 1]) = X \times \text{sk}_ n \Delta [n + 1]$ and that $i_{n!}(X \times \text{sk}_ n \Delta [n + 1]) = X \times i_{n !} \text{sk}_ n \Delta [n + 1]$. The first equality is obvious. For any (possibly truncated) simplicial object $W$ of $\mathcal{C}$ and any object $X$ of $\mathcal{C}$ denote temporarily $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W)$ the (possibly truncated) simplicial set $[n] \mapsto \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W_ n)$. From the definitions it follows that $\mathop{\mathrm{Mor}}\nolimits (U \times X, W) = \mathop{\mathrm{Mor}}\nolimits (U, \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W))$ for any (possibly truncated) simplicial set $U$. Hence

\begin{eqnarray*} \mathop{\mathrm{Mor}}\nolimits (X \times i_{n !} \text{sk}_ n \Delta [n + 1], W) & = & \mathop{\mathrm{Mor}}\nolimits (i_{n !} \text{sk}_ n \Delta [n + 1], \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W)) \\ & = & \mathop{\mathrm{Mor}}\nolimits (\text{sk}_ n \Delta [n + 1], \text{sk}_ n\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W)) \\ & = & \mathop{\mathrm{Mor}}\nolimits (X \times \text{sk}_ n \Delta [n + 1], \text{sk}_ nW) \\ & = & \mathop{\mathrm{Mor}}\nolimits (i_{n!}(X \times \text{sk}_ n \Delta [n + 1]), W). \end{eqnarray*}

This proves the second equality used, and ends the proof of the lemma. $\square$

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