$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 5.3.6.3. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a diagram of simplicial sets and suppose we are given morphisms of simplicial sets $A \xrightarrow {f} B \xrightarrow {g} \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, so that pullback along $f$ determines a map $\theta _ f: A \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}(\mathscr {F} ) \rightarrow B \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$. Then:
- $(1)$
If $f$ is right anodyne, then $\theta _{f}$ is right anodyne.
- $(2)$
If $f$ is inner anodyne, then $\theta _{f}$ is inner anodyne.
Proof.
We will give the proof of $(1)$; the proof of $(2)$ is similar. Let $S$ be the collection of all morphisms of simplicial sets $f: A \rightarrow B$ having the property that, for every morphism $g: B \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, the map $\theta _{f}$ is inner anodyne. It follows immediately from the definitions that $S$ is weakly saturated (in the sense of Definition 1.5.4.13). Consequently, to show that every right anodyne morphism belongs to $S$, it will suffice to prove that $S$ contained every inner horn inclusion $f: \Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, $0 < i \leq n$. Using Remark 5.3.2.3, we can reduce to the case where $\operatorname{\mathcal{C}}= [n]$ and $g: \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is the identity map. In this case, Remark 5.3.2.12 supplies a pushout diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \times \mathscr {F}(0) \ar [r] \ar [d] & \Lambda ^{n}_{i} \times _{ \Delta ^ n} \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [d] \\ \Delta ^ n \times \mathscr {F}(0) \ar [r] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ). } \]
It will therefore suffice to show that the left vertical map is right anodyne, which follows from Proposition 4.2.5.3.
$\square$