The Stacks project

Lemma 14.26.10. Let $\mathcal{C}$ be a category.

  1. If $a_ t, b_ t : X_ t \to Y_ t$, $t \in T$ are homotopic morphisms between simplicial objects of $\mathcal{C}$, then $\prod a_ t, \prod b_ t : \prod X_ t \to \prod Y_ t$ are homotopic morphisms between simplicial objects of $\mathcal{C}$, provided $\prod X_ t$ and $\prod Y_ t$ exist in $\text{Simp}(\mathcal{C})$.

  2. If $(X_ t, Y_ t)$, $t \in T$ are homotopy equivalent pairs of simplicial objects of $\mathcal{C}$, then $\prod X_ t$ and $\prod Y_ t$ are homotopy equivalent pairs of simplicial objects of $\mathcal{C}$, provided $\prod X_ t$ and $\prod Y_ t$ exist in $\text{Simp}(\mathcal{C})$.

Proof. If $h_ t = (h_{t, n , i})$ are homotopies connecting $a_ t$ and $b_ t$ (see Remark 14.26.4), then $h = (\prod _ t h_{t, n, i})$ is a homotopy connecting $\prod a_ t$ and $\prod b_ t$. This proves (1). Part (2) follows from part (1) and the definitions. $\square$


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