Definition 14.12.1. An $n$-truncated simplicial object of $\mathcal{C}$ is a contravariant functor from $\Delta _{\leq n}$ to $\mathcal{C}$. A morphism of $n$-truncated simplicial objects is a transformation of functors. We denote the category of $n$-truncated simplicial objects of $\mathcal{C}$ by the symbol $\text{Simp}_ n(\mathcal{C})$.
14.12 Truncated simplicial objects and skeleton functors
Let $\Delta _{\leq n}$ denote the full subcategory of $\Delta $ with objects $[0], [1], [2], \ldots , [n]$. Let $\mathcal{C}$ be a category.
Given a simplicial object $U$ of $\mathcal{C}$ the truncation $\text{sk}_ n U$ is the restriction of $U$ to the subcategory $\Delta _{\leq n}$. This defines a skeleton functor
\[ \text{sk}_ n : \text{Simp}(\mathcal{C}) \longrightarrow \text{Simp}_ n(\mathcal{C}) \]
from the category of simplicial objects of $\mathcal{C}$ to the category of $n$-truncated simplicial objects of $\mathcal{C}$. See Remark 14.21.6 to avoid possible confusion with other functors in the literature.
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Comment #1020 by correction_bot on