Kerodon

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Remark 7.5.2.8. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories. Arguing as in Remark 7.5.1.6, we can identify the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ with a simplicial subset of the product $\prod _{C \in \operatorname{\mathcal{C}}} \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ), \mathscr {F}(C) )$, whose $n$-simplices are collections of maps $\{ \sigma _{C}: \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \rightarrow \mathscr {F}(C) \} $ which satisfy the following pair of conditions:

$(\ast )$

For every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^{n} \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \ar [r]^-{\circ f} \ar [d]^{ \sigma _{C} } & \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/D} ) \ar [d]^{ \sigma _ D } \\ \mathscr {F}(C) \ar [r]^-{ \mathscr {F}(f) } & \mathscr {F}(D) } \]

is commutative.

$(\ast ')$

For every object $C \in \operatorname{\mathcal{C}}$ and every integer $0 \leq i \leq n$, the composite map

\[ \{ i\} \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \hookrightarrow \Delta ^ n \times \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{/C} ) \xrightarrow { \sigma _{C} } \mathscr {F}(C) \]

carries every morphism in the category $\operatorname{\mathcal{C}}_{/C}$ to an isomorphism in the $\infty $-category $\mathscr {F}(C)$.