Corollary 7.4.3.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and set $\operatorname{\mathcal{E}}^{0} = \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}^{0}$. Let $\kappa $ be an uncountable regular cardinal for which $U$ and $\operatorname{\mathcal{C}}$ are essentially $\kappa $-small. The following conditions are equivalent:
- $(1)$
The covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \kappa }$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$.
- $(2)$
The inclusion functor $\operatorname{\mathcal{E}}^{0} \hookrightarrow \operatorname{\mathcal{E}}$ is left cofinal.
Proof.
Fix an object $C \in \operatorname{\mathcal{C}}$ and set $\operatorname{\mathcal{C}}^{0}_{/C} = \operatorname{\mathcal{C}}^{0} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$. Note that the inclusion map $\operatorname{\mathcal{C}}^{0}_{/C} \rightarrow \operatorname{\mathcal{C}}_{/C}$ factors as a composition
\[ \operatorname{\mathcal{C}}^{0}_{/C} \hookrightarrow ( \operatorname{\mathcal{C}}^{0}_{ / C } )^{\triangleright } \xrightarrow {\lambda } \operatorname{\mathcal{C}}_{/C}, \]
where $\lambda $ carries the cone point $(\operatorname{\mathcal{C}}^{0}_{/C} )^{\triangleright }$ to the final object $\operatorname{id}_{C} \in \operatorname{\mathcal{C}}_{/C}$. In particular, $\lambda $ is right cofinal (Corollary 7.2.1.9). It follows that the induced map $\lambda _{\operatorname{\mathcal{E}}}: \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} (\operatorname{\mathcal{C}}^{0}_{/C})^{\triangleright } \rightarrow \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is also right cofinal (Proposition 7.2.3.12); in particular, $\lambda _{\operatorname{\mathcal{E}}}$ is a weak homotopy equivalence (Proposition 7.2.1.5). Combining this observation with Corollary 7.4.3.14, we see that the following conditions are equivalent:
- $(1_ C)$
The covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ at the object $C \in \operatorname{\mathcal{C}}$.
- $(1'_ C)$
The inclusion map $\iota _{C}: \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}^{0}_{/C} \hookrightarrow \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ is a weak homotopy equivalence.
Choose an object $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = C$ and set $\operatorname{\mathcal{E}}^{0}_{/X} = \operatorname{\mathcal{E}}^{0} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{E}}_{/X}$, so that we have a pullback diagram of simplicial sets
7.52
\begin{equation} \begin{gathered}\label{equation:left-fibration-Kan-extension-characterization} \xymatrix { \operatorname{\mathcal{E}}^{0}_{/X} \ar [r] \ar [d] & \operatorname{\mathcal{E}}_{/X} \ar [d] \\ \operatorname{\mathcal{E}}\times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}^{0}_{/C} \ar [r]^{\iota _{C}} & \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}. } \end{gathered} \end{equation}
Since $U$ is a left fibration, the vertical maps in this diagram are Kan fibrations; it follows that (7.52) is also a homotopy pullback square (Example 3.4.1.3). In particular, if condition $(1'_ C)$ is satisfied, then the inclusion map $\operatorname{\mathcal{E}}^{0}_{/X} \hookrightarrow \operatorname{\mathcal{E}}_{/X}$ is a weak homotopy equivalence (Corollary 3.4.1.5), so that the $\infty $-category $\operatorname{\mathcal{E}}^{0}_{/X}$ is weakly contractible. Conversely, if $\operatorname{\mathcal{E}}^{0}_{/X}$ is weakly contractible for every object $X \in \operatorname{\mathcal{C}}$ satisfying $U(X) = C$, then $\iota _{C}$ is a weak homotopy equivalence: this follows from the observation that every connected component of $\operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C}$ has nonempty intersection with the fiber $\operatorname{\mathcal{E}}_{C} = \operatorname{\mathcal{E}}\times _{\operatorname{\mathcal{C}}} \{ C\} $. It follows that $(1'_ C)$ can be reformulated as follows:
- $(2_ C)$
For every object $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = C$, the $\infty $-category $\operatorname{\mathcal{E}}^{0}_{/X}$ is weakly contractible.
The equivalence of $(1)$ and $(2)$ now follows from the equivalence $(1_ C) \Leftrightarrow (2_ C)$ by allowing the object $C \in \operatorname{\mathcal{C}}$ to vary (and applying Theorem 7.2.3.1).
$\square$