The Stacks project

Lemma 14.26.8. Let $\mathcal{C}$ be a category with finite coproducts. Let $U$ be a simplicial object of $\mathcal{C}$. Consider the maps $e_1, e_0 : U \to U \times \Delta [1]$, and $\pi : U \times \Delta [1] \to U$, see Lemma 14.13.3.

  1. We have $\pi \circ e_1 = \pi \circ e_0 = \text{id}_ U$, and

  2. The morphisms $\text{id}_{U \times \Delta [1]}$, and $e_0 \circ \pi $ are homotopic.

  3. The morphisms $\text{id}_{U \times \Delta [1]}$, and $e_1 \circ \pi $ are homotopic.

Proof. The first assertion is trivial. For the second, consider the map of simplicial sets $\Delta [1] \times \Delta [1] \longrightarrow \Delta [1]$ which in degree $n$ assigns to a pair $(\beta _1, \beta _2)$, $\beta _ i : [n] \to [1]$ the morphism $\beta : [n] \to [1]$ defined by the rule

\[ \beta (i) = \max \{ \beta _1(i), \beta _2(i)\} . \]

It is a morphism of simplicial sets, because the action $\Delta [1](\varphi ) : \Delta [1]_ n \to \Delta [1]_ m$ of $\varphi : [m] \to [n]$ is by precomposing. Clearly, using notation from Section 14.26, we have $\beta = \beta _1$ if $\beta _2 = \alpha ^ n_0$ and $\beta = \alpha ^ n_{n + 1}$ if $\beta _2 = \alpha ^ n_{n + 1}$. This implies easily that the induced morphism

\[ U \times \Delta [1] \times \Delta [1] \longrightarrow U \times \Delta [1] \]

of Lemma 14.13.3 is a homotopy from $\text{id}_{U \times \Delta [1]}$ to $e_0 \circ \pi $. Similarly for $e_1 \circ \pi $ (use minimum instead of maximum). $\square$

Comments (0)

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Your email address will not be published.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.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 019O. Beware of the difference between the letter 'O' and the digit '0'.