Kerodon

Example 7.5.2.8 (Homotopy Limits of Cores). Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories, and let $\mathscr {F}^{\simeq }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be the functor given on objects by the formula $\mathscr {F}^{\simeq }(C) = \mathscr {F}(C)^{\simeq }$. Then the inclusion map $\mathscr {F}^{\simeq } \hookrightarrow \mathscr {F}$ induces a monomorphism of simplicial sets $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}^{\simeq } ) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$, whose image is the core of the $\infty $-category $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ (see Example 5.3.3.21 and Remark 5.3.1.20). In other words, there is a canonical isomorphism of Kan complexes $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F}^{\simeq } ) \simeq \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F})^{\simeq }$.