Proposition 5.3.2.18. Let $\operatorname{\mathcal{C}}$ be a category and let $\alpha : \mathscr {F} \rightarrow \mathscr {G}$ be a levelwise weak homotopy equivalence between diagrams $\mathscr {F}, \mathscr {G}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$. Then the induced map $ \underset { \longrightarrow }{\mathrm{holim}}(\alpha ): \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} )$ is a weak homotopy equivalence of simplicial sets.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Proposition 3.4.2.16, it will suffice to show that for every $n$-simplex $\Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the induced map $\Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow \Delta ^{n} \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {G})$ is a weak homotopy equivalence. Using Remark 5.3.2.3, we are reduced to proving Proposition 5.3.2.18 in the special case where $\operatorname{\mathcal{C}}$ is the linearly ordered set $[n] = \{ 0 < 1 < \cdots < n \} $. We now proceed by induction on $n$. If $n = 0$, the desired result follows immediately from Example 5.3.2.2. Let us therefore assume that $n > 0$. Let $\mathscr {F}'$ denote the restriction of $\mathscr {F}$ to the full subcategory $\{ 1 < 2 < \cdots < n \} $ and define $\mathscr {G}'$ similarly. The natural transformation $\alpha $ determines a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Delta ^ n \times \mathscr {F}(0) \ar [d] & \operatorname{N}_{\bullet }( \{ 1 < \cdots < n \} ) \times \mathscr {F}(0) \ar [r] \ar [d] \ar [l] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}' ) \ar [d] \\ \Delta ^ n \times \mathscr {G}(0) & \operatorname{N}_{\bullet }( \{ 1 < \cdots < n \} ) \times \mathscr {G}(0) \ar [l] \ar [r] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G}' ), } \]
where the left horizontal maps are monomorphisms, the right vertical map is a weak homotopy equivalence by virtue of our inductive hypothesis, and the other vertical maps are weak homotopy equivalences by virtue of our assumption on $\alpha $. The desired result now follows by combining Corollary 3.4.2.14 with Remark 5.3.2.12. $\square$