Kerodon

Variant 9.2.7.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ and $T$ be collections of morphisms of $\operatorname{\mathcal{C}}$. We say that $S$ is left orthogonal to $T$ if every morphism $f \in S$ is weakly left orthogonal to every morphism $g \in T$. In this case, we also say that $T$ is right orthogonal to $S$. In the special case where $S = \{ f\} $ is a singleton, we abbreviate this condition by saying that $f$ is left orthogonal to $T$, or $T$ is right orthogonal to $f$. In the special case $T = \{ g\} $ is a singleton, we abbreviate this condition by saying that $g$ is right orthogonal to $S$, or $S$ is left orthogonal to $g$.