The Stacks project

Remark 14.19.8. The construction of Lemma 14.19.6 above in the case of simplicial sets is the following. Given an $n$-truncated simplicial set $U$, we make a canonical $(n + 1)$-truncated simplicial set $\tilde U$ as follows. We add a set of $(n + 1)$-simplices $U_{n + 1}$ by the formula of the lemma. Namely, an element of $U_{n + 1}$ is a numbered collection of $(f_0, \ldots , f_{n + 1})$ of $n$-simplices, with the property that they glue as they would in a $(n + 1)$-simplex. In other words, the $i$th face of $f_ j$ is the $(j-1)$st face of $f_ i$ for $i < j$. Geometrically it is obvious how to define the face and degeneracy maps for $\tilde U$. If $V$ is an $(n + 1)$-truncated simplicial set, then its $(n + 1)$-simplices give rise to compatible collections of $n$-simplices $(f_0, \ldots , f_{n + 1})$ with $f_ i \in V_ n$. Hence there is a natural map $\mathop{\mathrm{Mor}}\nolimits (\text{sk}_ nV, U) \to \mathop{\mathrm{Mor}}\nolimits (V, \tilde U)$ which is inverse to the canonical restriction mapping the other way.

Also, it is enough to do the combinatorics of the construction in the case of truncated simplicial sets. Namely, for any object $T$ of the category $\mathcal{C}$, and any $n$-truncated simplicial object $U$ of $\mathcal{C}$ we can consider the $n$-truncated simplicial set $\mathop{\mathrm{Mor}}\nolimits (T, U)$. We may apply the construction to this, and take its set of $(n + 1)$-simplices, and require this to be representable. This is a good way to think about the result of Lemma 14.19.6.