Kerodon

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

Remark 9.6.6.21. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. Then $f$ is weakly left orthogonal to $g$ if and only if, for every lifting problem $\sigma :$

\[ \xymatrix@C =40pt@R=40pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur] & Y, } \]

the solution space $\operatorname{Sol}( \sigma )$ is nonempty. In particular, if $f$ is left orthogonal to $g$, then it is weakly left orthogonal to $g$. Beware that the converse is false in general (Exercise 9.6.6.22).