Kerodon

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

Proposition 7.1.2.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a morphism with restriction $f = \overline{f}|_{K}$. The following conditions are equivalent:

$(1)$

The morphism $\overline{f}$ is a limit diagram (Definition 7.1.2.4).

$(2)$

The restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a trivial Kan fibration.

$(3)$

The restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is an equivalence of $\infty $-categories.

$(4)$

For every object $X \in \operatorname{\mathcal{C}}$, the restriction map $\{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \{ X\} \times \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f}$ is a homotopy equivalence of Kan complexes.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows from Proposition 4.6.6.11. Note that the restriction map $\operatorname{\mathcal{C}}_{ / \overline{f} } \rightarrow \operatorname{\mathcal{C}}_{/f}$ is a right fibration of $\infty $-categories (Corollary 4.3.6.11), and therefore an isofibration (Example 4.4.1.10). The equivalence $(2) \Leftrightarrow (3)$ now follows from Proposition 4.5.5.20, and the equivalence $(3) \Leftrightarrow (4)$ follows from Corollary 5.1.5.4. $\square$