Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Definition 5.1.3.1. Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr]_{q_ X} & & Y \ar [dl]^{q_ Y} \\ & S, & } \]

so that we can regard $f$ as a vertex of the simplicial set $\operatorname{Fun}_{/S}(X,Y)$ of Construction 3.1.3.7. We say that $f$ is a homotopy equivalence relative to $S$ if there exists a morphism $g \in \operatorname{Fun}_{/S}(Y,X)$ such that $g \circ f$ and $\operatorname{id}_{X}$ belong to the same connected component of $\operatorname{Fun}_{/S}(X,X)$, and $f \circ g$ and $\operatorname{id}_{Y}$ belong to the same connected component of $\operatorname{Fun}_{/S}(Y,Y)$. In this case, we will refer to $g$ as a homotopy inverse of $f$ relative to $S$.