The Stacks project

Definition 14.26.6. Let $U$ and $V$ be two simplicial objects of a category $\mathcal{C}$. We say a morphism $a : U \to V$ is a homotopy equivalence if there exists a morphism $b : V \to U$ such that $a \circ b$ is homotopic to $\text{id}_ V$ and $b \circ a$ is homotopic to $\text{id}_ U$. We say $U$ and $V$ are homotopy equivalent if there exists a homotopy equivalence $a : U \to V$.