Kerodon

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

Example 4.6.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors between ordinary categories, and let $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})$ denote the oriented fiber product of Notation 2.1.4.19. Since the nerve construction is compatible with the formation of inverse limits and functor categories, we have a canonical isomorphism of simplicial sets

\[ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}) \simeq ( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}})} \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) ). \]

Consequently, Definition 4.6.4.1 can be viewed as a generalization of the classical oriented fiber product.