Kerodon

Warning 3.1.6.2. Let $f: X \rightarrow Y$ be a morphism of simplicial sets. Many authors refer to a morphism $g: Y \rightarrow X$ as 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. However, when $X$ and $Y$ are $\infty $-categories, it is natural to consider a different (and more restrictive) notion of homotopy inverse, which requires that $g \circ f$ and $f \circ g$ be isomorphic to $\operatorname{id}_{X}$ and $\operatorname{id}_{Y}$ as objects of the $\infty $-categories $\operatorname{Fun}(X,X)$ and $\operatorname{Fun}(Y,Y)$, respectively (see Definition 4.5.1.10 and Warning 4.5.1.14). For this reason, we will use the term simplicial homotopy inverse in the setting of Definition 3.1.6.1 (unless $X$ and $Y$ are Kan complexes, in which case the distinction disappears).