$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.5.5.7. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a functor. The following conditions are equivalent:
- $(1)$
The functor $\overline{\mathscr {F}}$ is a categorical limit diagram, in the sense of Definition 7.5.5.1.
- $(2)$
For every simplicial set $K$, the functor
\[ \overline{\mathscr {F}}^{K}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}\quad \quad C \mapsto \operatorname{Fun}( K, \overline{\mathscr {F}}(C) ) \]
is a categorical limit diagram.
- $(3)$
For every simplicial set $K$, the functor
\[ (\overline{\mathscr {F}}^{K})^{\simeq }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}\quad \quad C \mapsto \operatorname{Fun}( K, \overline{\mathscr {F}}(C) )^{\simeq } \]
is a homotopy limit diagram.
- $(4)$
The functor $( \overline{\mathscr {F}}^{\Delta ^1} )^{\simeq }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$ is a homotopy limit diagram.
Proof.
The implication $(1) \Rightarrow (2)$ follows from Remarks 7.5.2.3 and 4.5.1.16, the implication $(2) \Rightarrow (3)$ from Remark 7.5.5.5, and the implication $(3) \Rightarrow (4)$ is immediate. Set $\mathscr {F} = \overline{\mathscr {F}}|_{\operatorname{\mathcal{C}}}$, and let ${\bf 0}$ denote the initial object of $\operatorname{\mathcal{C}}^{\triangleleft }$. Using Remark 7.5.2.3 and Example 7.5.2.8, we see that condition $(4)$ is equivalent to the requirement that the map $\overline{ \mathscr {F} }( {\bf 0} ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ induces a homotopy equivalence of Kan complexes
\[ \operatorname{Fun}( \Delta ^1, \overline{ \mathscr {F} }( {\bf 0} ))^{\simeq } \rightarrow \operatorname{Fun}( \Delta ^1, \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} ))^{\simeq } \simeq \underset {\longleftarrow }{\mathrm{holim}}( ( \overline{\mathscr {F}}^{\Delta ^1} )^{\simeq } ). \]
The implication $(4) \Rightarrow (1)$ now follows from Theorem 4.5.7.1.
$\square$