Section 14.12 (017Z): Truncated simplicial objects and skeleton functors

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.

Comments (1)

Comment #1020 by correction_bot on September 09, 2014 at 04:24

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.

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14.12 Truncated simplicial objects and skeleton functors

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