The Stacks project

Remark 14.28.2. Let $\mathcal{C}$ be any category (no assumptions whatsoever). Let $U$ and $V$ be cosimplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be morphisms of cosimplicial objects of $\mathcal{C}$. A homotopy from $a$ to $b$ is given by morphisms $h_{n, \alpha } : U_ n \to V_ n$, for $n \geq 0$, $\alpha \in \Delta [1]_ n$ satisfying ( for all morphisms $f$ of $\Delta $ and such that $a_ n = h_{n, 0 : [n] \to [1]}$ and $b_ n = h_{n, 1 : [n] \to [1]}$ for all $n \geq 0$. As in Definition 14.28.1 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. This new notion is the same as the old one in case finite products exist. We deduce in particular that functors preserve the original notion whenever both categories have finite products.

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