Kerodon

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

Remark 2.3.5.9. Let $\operatorname{\mathcal{E}}$ be a category which admits fiber products. Combining Corollary 2.3.5.8 with Example 2.3.5.4, we see that $n$-simplices of the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Corr}(\operatorname{\mathcal{E}})^{\operatorname{c}} )$ can be identified with diagrams

\[ \{ (i,j) \in [n] \times [n]^{\operatorname{op}}: i \leq j \} \rightarrow \operatorname{\mathcal{E}}, \]

which we can represent graphically as

\[ \xymatrix { & & & X_{0,n} \ar [dl] \ar [dr] & & & \\ & & \cdots \ar [dl] \ar [dr] & & \cdots \ar [dl] \ar [dr] & & \\ & X_{0,1} \ar [dl] \ar [dr] & & \cdots \ar [dl]\ar [dr] & & X_{n-1,n} \ar [dl] \ar [dr] & \\ X_{0,0} & & X_{1,1} & \cdots & X_{n-1,n-1} & & X_{n,n}. } \]

In ยง, we will use this description to extend the definition of $\operatorname{Corr}(\operatorname{\mathcal{E}})$ to the case where $\operatorname{\mathcal{E}}$ is an $\infty $-category.