The Stacks project

Remark 14.26.4. Let $\mathcal{C}$ be any category (no assumptions whatsoever). Let $U$ and $V$ be simplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be morphisms of simplicial objects of $\mathcal{C}$. A homotopy from $a$ to $b$ is given by 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. As in Definition 14.26.1 we say the morphisms $a$ and $b$ are homotopic if there exists a sequence of morphisms $a = a_0, a_1, \ldots , a_ n = b$ from $U$ to $V$ such that for each $i = 1, \ldots , n$ there either exists a homotopy from $a_{i - 1}$ to $a_ i$ or there exists a homotopy from $a_ i$ to $a_{i - 1}$. Clearly, if $F : \mathcal{C} \to \mathcal{C}'$ is any functor and $\{ h_{n, i}\} $ is a homotopy from $a$ to $b$, then $\{ F(h_{n, i})\} $ is a homotopy from $F(a)$ to $F(b)$. Similarly, if $a$ and $b$ are homotopic, then $F(a)$ and $F(b)$ are homotopic. 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 original 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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