Kerodon

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

Remark 7.3.6.8. In the situation of Proposition 7.3.6.7, the horizontal maps in the diagram (7.27) are Kan fibrations (Corollary 4.1.4.2 and Proposition 4.6.1.20). Consequently, the diagram (7.27) is a homotopy pullback square if and only if the induced map

\[ \xymatrix@C =50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G ) \ar [d]^{\theta } \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) }( F_0, G_0) \times _{ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) }( UF_0 , UG_0) } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) }( UF, UG)} \]

is a homotopy equivalence (Example 3.4.1.3). Writing $\operatorname{\mathcal{M}}$ for the fiber product

\[ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]

and $V: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{M}}$ for the functor given by $V(H) = ( H|_{\operatorname{\mathcal{C}}^{0}}, U \circ H)$, we can identify $\theta $ with the map $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{M}}}( V(F), V(G) )$ determined by $V$. We can therefore restate Proposition 7.3.6.7 as follows:

  • If the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, then it is $V$-initial when viewed as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.

  • If the functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, then it is $V$-final when viewed as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.