Kerodon

Warning 7.5.4.3. For every diagram of Kan complexes $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Kan}$, the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ of Construction 7.5.1.1 is well-defined. However, one cannot always extend $\mathscr {F}$ to a homotopy limit diagram $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$ (see Warning 3.4.1.8). This is possible only if the tautological map $\varprojlim (\mathscr {F}) \hookrightarrow \underset {\longleftarrow }{\mathrm{holim}}(\mathscr {F})$ has a left homotopy inverse. However, we can always choose a levelwise homotopy equivalence $\alpha : \mathscr {F} \hookrightarrow \mathscr {G}$, where $\mathscr {G}$ is an isofibrant diagram of Kan complexes (Variant 7.5.3.6). We can then extend $\mathscr {G}$ can be extended to a limit diagram $\overline{ \mathscr {G} }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$, which is also a homotopy limit diagram (Example 7.5.4.2). Moreover, if take $\mathscr {G} = \mathscr {F}^{+}$ to be the isofibrant replacement of Construction 7.5.3.3, then $\overline{\mathscr {G}}$ carries the initial object of $\operatorname{\mathcal{C}}^{\triangleleft }$ to the homotopy limit $ \underset {\longleftarrow }{\mathrm{holim}}( \mathscr {F} )$ (Proposition 7.5.3.7).