The Stacks project

Definition 14.26.1. Let $\mathcal{C}$ be a category having finite coproducts. Suppose that $U$ and $V$ are two simplicial objects of $\mathcal{C}$. Let $a, b : U \to V$ be two morphisms.

1. We say a morphism

   $$h : U \times \Delta [1] \longrightarrow V$$

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

2. We say the morphisms $a$ and $b$ are homotopic or are in the same homotopy class 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 $a_{i - 1}$.