Construction 7.5.1.1 (Homotopy Limits of Kan Complexes). Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$ be a diagram of Kan complexes indexed by $\operatorname{\mathcal{C}}$, and $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}})$ denote the weighted nerve of $\mathscr {F}$ (Definition 5.3.3.1). We define
to be the simplicial set which parametrizes sections of the projection map $\operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$. We will refer to $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ as the homotopy limit of the diagram $\mathscr {F}$.