Example 7.5.2.16 (03AM)

Example 7.5.2.16. Let $[1]$ denote the linearly ordered set $\{ 0 < 1 \} $ and let $\mathscr {F}: [1] \rightarrow \operatorname{QCat}$ be a diagram, which we identify with a functor of $\infty $-categories $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Then the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ of Construction 7.5.1.1 can be identified with the homotopy fiber product

\[ \operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}= \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}( \{ 0\} , \operatorname{\mathcal{D}}) } \operatorname{Isom}(\operatorname{\mathcal{D}}) \]

of Construction 4.5.2.1. Under this identification, the comparison morphism $\varprojlim (\mathscr {F}) \rightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ of Remark 7.5.2.12 corresponds to the monomorphism

\[ \operatorname{\mathcal{C}}\simeq \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}\hookrightarrow \operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}}^{\mathrm{h}} \operatorname{\mathcal{D}} \]

of Proposition 3.4.0.7. This morphism is usually not an isomorphism of simplicial sets, though it is always an equivalence of $\infty $-categories (Proposition 7.5.2.13).