Definition Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. We will say that a morphism $g: Y_{} \rightarrow X_{}$ is a homotopy inverse to $f$ if the compositions $g \circ f$ and $f \circ g$ are homotopic to the identity morphisms $\operatorname{id}_{X_{}}$ and $\operatorname{id}_{ Y_{} }$, respectively (in the sense of Definition We say that $f: X_{} \rightarrow Y_{}$ is a homotopy equivalence if it admits a homotopy inverse $g$.