The Stacks project

Definition 14.28.1. Let $\mathcal{C}$ be a category having finite products. Let $U$ and $V$ be two cosimplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be two morphisms of cosimplicial objects of $\mathcal{C}$.

1. We say a morphism

   $$h : U \longrightarrow \mathop{\mathrm{Hom}}\nolimits (\Delta [1], V)$$

   such that $a = e_0 \circ h$ and $b = e_1 \circ h$ is a homotopy from $a$ to $b$.

2. We say $a$ and $b$ are homotopic or are in the same homotopy class if there exists a sequence $a = a_0, a_1, \ldots , a_ n = b$ of morphisms from $U$ to $V$ such that for each $i = 1, \ldots , n$ there either exists a homotopy from $a_ i$ to $a_{i - 1}$ or there exists a homotopy from $a_{i - 1}$ to $a_ i$.