Kerodon

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

Remark 9.2.5.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a morphism $g: X \rightarrow Y$, which we identify with an object $\widetilde{X}$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$. Let $S$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $\widetilde{S}$ denote its inverse image in $\operatorname{\mathcal{C}}_{/Y}$. The following conditions are equivalent:

  • The morphism $g$ is weakly right orthogonal to $S$ (in the sense of Variant 9.2.5.11).

  • The object $\widetilde{X}$ is weakly $\widetilde{S}$-local (in the sense of Definition 9.2.3.5).