Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Example 5.2.4.8. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors between categories. Then the relative join $\operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) \star _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ can be identified with the nerve of the category

\[ \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}= (\operatorname{\mathcal{C}}\star \operatorname{\mathcal{D}}) \times _{ ( \operatorname{\mathcal{E}}\star \operatorname{\mathcal{E}}) } ( [1] \times \operatorname{\mathcal{E}}), \]

which can be described more concretely as follows:

  • The set of objects $\operatorname{Ob}( \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})$ is the disjoint union of $\operatorname{Ob}(\operatorname{\mathcal{C}})$ with $\operatorname{Ob}(\operatorname{\mathcal{D}})$.

  • For every pair of objects $X,Y \in \operatorname{Ob}( \operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}})$, we have

    \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}\star _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}}( X, Y) = \begin{cases} \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) & \textnormal{ if $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{C}})$} \\ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( X,Y) & \textnormal{ if $X,Y \in \operatorname{Ob}(\operatorname{\mathcal{D}})$} \\ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( F(X), G(Y) ) & \textnormal{ if $X \in \operatorname{Ob}(\operatorname{\mathcal{C}})$, $Y \in \operatorname{Ob}(\operatorname{\mathcal{D}})$} \\ \emptyset & \textnormal{ if $X \in \operatorname{Ob}(\operatorname{\mathcal{D}})$, $Y \in \operatorname{Ob}(\operatorname{\mathcal{C}}).$} \end{cases} \]