# Kerodon

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

Proposition 10.2.5.9. Let $\operatorname{\mathcal{C}}$ be a regular $\infty$-category. Then, for every object $Z \in \operatorname{\mathcal{C}}$, the slice $\infty$-category $\operatorname{\mathcal{C}}_{/Z}$ is regular.

Proof. It follows from Remark 7.1.2.11 that the $\infty$-category $\operatorname{\mathcal{C}}_{/Z}$ admits finite limits. By virtue of Corollary 10.2.5.5, it will suffice to show that every morphism $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ can be realized as the composition of a universal quotient morphism $\widetilde{X} \twoheadrightarrow \widetilde{Y}_0$ with a monomorphism $\widetilde{Y}_0 \hookrightarrow \widetilde{Y}$. Let $f: X \rightarrow Y$ denote the image of $\widetilde{f}$ in the $\infty$-category $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is regular, we can choose a $2$-simplex $\sigma :$

$\xymatrix@R =50pt@C=50pt{ & Y_0 \ar [dr]^{i} & \\ X \ar [ur]^{q} \ar [rr]^{f} & & Y }$

of $\operatorname{\mathcal{C}}$, where $q$ is a universal quotient morphism and $i$ is a monomorphism. The inclusion map $\operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \hookrightarrow \Delta ^2$ is right anodyne (Lemma 4.3.7.8), so we can lift $\sigma$ to a $2$-simplex

$\xymatrix@R =50pt@C=50pt{ & \widetilde{Y}_0 \ar [dr]^{ \widetilde{i} } & \\ \widetilde{X} \ar [ur]^{ \widetilde{q} } \ar [rr]^{ \widetilde{f} } & & \widetilde{Y} }$

in the $\infty$-category $\operatorname{\mathcal{C}}_{/Z}$. We conclude by observing that $\widetilde{i}$ is a monomorphism (Remark 9.2.4.23) and $\widetilde{q}$ is a universal quotient morphism (Proposition 10.2.4.15). $\square$