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 $, and $\pi : U \times \Delta  \to U$, see Lemma 14.13.3.

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

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

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

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

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

It is a morphism of simplicial sets, because the action $\Delta (\varphi ) : \Delta _ n \to \Delta _ m$ 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  \times \Delta  \longrightarrow U \times \Delta $

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

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