The Stacks project

Remark 14.26.4. Let $\mathcal{C}$ be any category (no assumptions whatsoever). We say that a pair of morphisms $a, b : U \to V$ of simplicial objects are homotopic if there exist morphisms1 $h_{n, i} : U_ n \to V_ n$, for $n \geq 0$, $i = 0, \ldots , n + 1$ satisfying the relations of Lemma 14.26.2 (potentially with the roles of $a$ and $b$ switched). This is a “better” definition, because it applies to any category. Also it has the following property: if $F : \mathcal{C} \to \mathcal{C}'$ is any functor then $a$ homotopic to $b$ implies trivially that $F(a)$ is homotopic to $F(b)$. Since the lemma says that the newer notion is the same as the old one in case finite coproduct exist, we deduce in particular that functors preserve the old notion whenever both categories have finite coproducts.

[1] In the literature, often the maps $h_{n + 1, i} \circ s_ i : U_ n \to V_{n + 1}$ are used instead of the maps $h_{n, i}$. Of course the relations these maps satisfy are different from the ones in Lemma 14.26.2.

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