## 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.

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 $\text{Simp}_ n(\mathcal{C})$.

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.

Comment #1020 by correction_bot on

Hah, doesn't "skeleton functor" sound better than "skelet functor," and similarly for the co's? If you want to keep it "skelet," then maybe the title of section 14.19 should be changed.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).