Kerodon

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

Remark 9.5.0.5. Let $\operatorname{\mathcal{C}}$ be a category which admits pushouts. Stated more informally, Proposition None asserts that $n$-simplices of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Corr}(\operatorname{\mathcal{C}}) )$ can be identified with commutative diagrams

\[ \xymatrix@R =25pt@C=25pt{ & & & X_{0,n} \ar [dl] \ar [dr] & & & \\ & & \cdots \ar [dl] \ar [dr] & & \cdots \ar [dl] \ar [dr] & & \\ & X_{0,1} \ar [dl] \ar [dr] & & \cdots \ar [dl]\ar [dr] & & X_{n-1,n} \ar [dl] \ar [dr] & \\ X_{0,0} & & X_{1,1} & \cdots & X_{n-1,n-1} & & X_{n,n} } \]

in the category $\operatorname{\mathcal{C}}$.