Kerodon

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

Corollary 10.2.2.7. Let $\operatorname{\mathcal{C}}$ be a category which admits fiber products and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $f$ is a quotient morphism in the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 10.2.2.1).

$(2)$

The morphism $f$ is a regular epimorphism: that is, it exhibits $Y$ as a coequalizer of the projection maps $X \times _{Y} X \rightrightarrows X$ (Definition 10.2.0.2).