The Stacks project

Lemma 14.21.1. Let $\mathcal{C}$ be a category which has finite colimits. The functors $i_{m!}$ exist for all $m$. Let $U$ be an $m$-truncated simplicial object of $\mathcal{C}$. The simplicial object $i_{m!}U$ is described by the formula

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

and for $\varphi : [n] \to [n']$ the map $i_{m!}U(\varphi )$ comes from the identification $U(n) \circ \underline{\varphi } = U(n')$ above via Categories, Lemma 4.14.8.

Proof. In this proof we denote $i_{m!}U$ the simplicial object whose $n$th term is given by the displayed formula of the lemma. We will show it satisfies the adjointness property.

Let $V$ be a simplicial object of $\mathcal{C}$. Let $\gamma : U \to \text{sk}_ mV$ be given. A morphism

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

is given by a compatible system of morphisms $f_\alpha : U_ k \to T$ where $\alpha : [n] \to [k]$ with $k \leq m$. Certainly, we have such a system of morphisms by taking the compositions

\[ U_ k \xrightarrow {\gamma _ k} V_ k \xrightarrow {V(\alpha )} V_ n. \]

Hence we get an induced morphism $(i_{m!}U)_ n \to V_ n$. We leave it to the reader to see that these form a morphism of simplicial objects $\gamma ' : i_{m!}U \to V$.

Conversely, given a morphism $\gamma ' : i_{m!}U \to V$ we obtain a morphism $\gamma : U \to \text{sk}_ m V$ by setting $\gamma _ i : U_ i \to V_ i$ equal to the composition

\[ U_ i \xrightarrow {\text{id}_{[i]}} \mathop{\mathrm{colim}}\nolimits _{([i]/\Delta )_{\leq m}^{opp}} U(i) \xrightarrow {\gamma '_ i} V_ i \]

for $0 \leq i \leq n$. We leave it to the reader to see that this is the inverse of the construction above. $\square$

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