Lemma 113.10.1. Assumptions and notation as in Simplicial, Lemma 14.32.1. There exists a section $g : U \to V$ to the morphism $f$ and the composition $g \circ f$ is homotopy equivalent to the identity on $V$. In particular, the morphism $f$ is a homotopy equivalence.

## 113.10 Simplicial methods

**Proof.**
Immediate from Simplicial, Lemmas 14.32.1 and 14.30.8.
$\square$

Lemma 113.10.2. Let $\mathcal{C}$ be a category with finite coproducts and finite limits. Let $X$ be an object of $\mathcal{C}$. Let $k \geq 0$. The canonical map

is an isomorphism.

**Proof.**
For any simplicial object $V$ we have

The first equality by the adjointness of $\text{sk}$ and $\text{cosk}$, the second equality by the adjointness of $i_{1!}$ and $\text{sk}_1$, and the first equality by Simplicial, Definition 14.17.1 where the last $X$ denotes the constant simplicial object with value $X$. By Simplicial, Lemma 14.20.2 an element in this set depends only on the terms of degree $0$ and $1$ of $i_{1!} \text{sk}_1 V \times \Delta [k]$. These agree with the degree $0$ and $1$ terms of $V \times \Delta [k]$, see Simplicial, Lemma 14.21.3. Thus the set above is equal to $\mathop{Mor}\nolimits (V \times \Delta [k], X) = \mathop{Mor}\nolimits (V, \mathop{\mathrm{Hom}}\nolimits (\Delta [k], X))$. $\square$

Lemma 113.10.3. Let $\mathcal{C}$ be a category. Let $X$ be an object of $\mathcal{C}$ such that the self products $X \times \ldots \times X$ exist. Let $k \geq 0$ and let $C[k]$ be as in Simplicial, Example 14.5.6. With notation as in Simplicial, Lemma 14.15.2 the canonical map

is identified with the map

which is the projection onto the factors where $\alpha $ is a constant map.

**Proof.**
This is shown in the proof of Hypercoverings, Lemma 25.7.3.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)