Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Lemma 1.4.6.9 (Joyal).

$(a)$

For every monomorphism of simplicial sets $i: A_{\bullet } \hookrightarrow B_{\bullet }$, the induced map

\[ (B_{\bullet } \times \Lambda ^2_1) \coprod _{A_{\bullet } \times \Lambda ^2_1 } (A_{\bullet } \times \Delta ^2) \subseteq B_{\bullet } \times \Delta ^2 \]

is inner anodyne.

$(b)$

The collection of inner anodyne morphisms is generated (as a weakly saturated class) by the inclusion maps

\[ (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \subseteq \Delta ^ m \times \Delta ^2 \]

for $m \geq 0$.

Proof. Let $T$ be the weakly saturated class of morphisms generated by all inclusions of the form

\[ (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \subseteq \Delta ^ m \times \Delta ^2, \]

and let $S$ be the collection of all morphisms of simplicial sets $A_{\bullet } \rightarrow B_{\bullet }$ for which the map

\[ (B_{\bullet } \times \Lambda ^2_1) \coprod _{A_{\bullet } \times \Lambda ^2_1 } (A_{\bullet } \times \Delta ^2) \subseteq B_{\bullet } \times \Delta ^2 \]

belongs to $T$. By construction, $S$ contains all inclusions of the form $\operatorname{\partial \Delta }^ m \hookrightarrow \Delta ^ m$. Moreover, since $T$ is weakly saturated, the class $S$ is also weakly saturated. It follows that every monomorphism of simplicial sets belongs to $S$ (Proposition 1.4.5.12). Consequently, to prove Lemma 1.4.6.9, it will suffice to show that $T$ coincides with the class of inner anodyne morphisms of $\operatorname{Set_{\Delta }}$. We first show that every inner anodyne morphism belongs to $T$. Since $T$ is weakly saturated, we are reduced to showing that every inner horn inclusion $f: \Lambda ^{n}_ i \hookrightarrow \Delta ^ n$ belongs to $T$. Since $f$ belongs to $S$, the monomorphism

\[ \overline{f}: (\Delta ^ n \times \Lambda ^2_1) \coprod _{ \Lambda ^ n_ i \times \Lambda ^2_1 } ( \Lambda ^ n_ i \times \Delta ^2) \subseteq \Delta ^ n \times \Delta ^2. \]

belongs to $T$. We conclude by observing that the morphism $f$ is a retract of $\overline{f}$. More precisely, we have a commutative diagram of simplicial sets

\[ \xymatrix@C =40pt@R=40pt{ \Lambda ^ n_ i \ar [r] \ar [d]^{f} & (\Delta ^ n \times \Lambda ^2_1) \coprod _{ \Lambda ^ n_ i \times \Lambda ^2_1 } ( \Lambda ^ n_ i \times \Delta ^2) \ar [r] \ar [d]^{\overline{f}} & \Lambda ^ n_ i \ar [d]^{f} \\ \Delta ^ n \ar [r]^{s} & \Delta ^ n \times \Delta ^2 \ar [r]^{r} & \Delta ^ n, } \]

where the maps $s$ and $r$ are given on vertices by the formulae

\[ s(j) = \begin{cases} (j,0) & \text{if } j < i \\ (j,1) & \text{if } j = i \\ (j,2) & \text{if } j > i \end{cases} \]

\[ r(j,k) = \begin{cases} j & \text{if } j < i, k=0 \\ j & \text{if } j > i, k = 2 \\ i & \text{otherwise.} \end{cases} \]

We now show that every morphism of $T$ is inner anodyne. Since the collection of inner anodyne morphisms is weakly saturated, it will suffice to show that the inclusion map

\[ (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2) \subseteq \Delta ^ m \times \Delta ^2 \]

is inner anodyne for each $m \geq 0$. For each $0 \leq i \leq j < m$, we let $\sigma _{ij}$ denote the $(m+1)$-simplex of $\Delta ^ m \times \Delta ^2$ given by the map of partially ordered sets

\[ f_{ij}: [m+1] \rightarrow [m] \times [2] \]

\[ f_{ij}(k) = \begin{cases} (k,0) & \text{if } 0 \leq k \leq i \\ (k-1, 1) & \text{if } i+1 \leq k \leq j+1 \\ (k-1, 2) & \text{if } j+2 \leq k \leq m+1. \end{cases} \]

For each $0 \leq i \leq j \leq m$, we let $\tau _{ij}$ denote the $(m+2)$-simplex of $\Delta ^ m \times \Delta ^2$ given by the map of partially ordered sets

\[ g_{ij}: [m+2] \rightarrow [m] \times [2] \]

\[ g_{ij}(k) = \begin{cases} (k,0) & \text{if } 0 \leq k \leq i \\ (k-1, 1) & \text{if } i+1 \leq k \leq j+1 \\ (k-2, 2) & \text{if } j+2 \leq k \leq m+2. \end{cases} \]

We will regard each $\sigma _{ij}$ and $\tau _{ij}$ as a simplicial subset of $\Delta ^ m \times \Delta ^2$.

Set $X(0) = (\Delta ^ m \times \Lambda ^2_1) \coprod _{ \operatorname{\partial \Delta }^ m \times \Lambda ^2_1 } (\operatorname{\partial }\Delta ^ m \times \Delta ^2)$. For $0 \leq j < m$, we let

\[ X(j+1) = X(j) \cup \sigma _{0j} \cup \cdots \cup \sigma _{jj}. \]

We have a chain of inclusions

\[ X(j) \subseteq X(j) \cup \sigma _{0j} \subseteq \cdots \subseteq X(j) \cup \sigma _{0j} \cup \cdots \cup \sigma _{jj} = X(j+1). \]

Each of these inclusions fits into a pushout diagram

\[ \xymatrix { \Lambda ^{m+1}_{ i+1} \ar [r] \ar [d] & X(j) \cup \sigma _{0j} \cup \cdots \cup \sigma _{(i-1)j} \ar [d] \\ \sigma _{ij} \ar [r] & X(j) \cup \sigma _{0j} \cup \cdots \cup \sigma _{ij}, } \]

and is therefore inner anodyne. Set $Y(0) = X(m)$, so that the inclusion $X(0) \subseteq Y(0)$ is inner anodyne. We now set $Y(j+1) = Y(j) \cup \tau _{0j} \cup \cdots \cup \tau _{jj}$ for $0 \leq j \leq m$. As before, we have a chain of inclusions

\[ Y(j) \subseteq Y(j) \cup \tau _{0j} \subseteq \cdots \subseteq Y(j) \cup \tau _{0j} \cup \cdots \cup \tau _{jj} = Y(j+1), \]

each of which fits into a pushout diagram

\[ \xymatrix { \Lambda ^{m+2}_{i+1} \ar [r] \ar [d] & Y(j) \cup \tau _{0j} \cup \cdots \cup \tau _{(i-1)j} \ar [d] \\ \tau _{ij} \ar [r] & Y(j) \cup \tau _{0j} \cup \cdots \cup \tau _{ij}, } \]

and is therefore inner andoyne. It follows that each inclusion $Y(j) \subseteq Y(j+1)$ is inner anodyne. Since the collection of inner anodyne morphisms is closed under composition, we conclude that the inclusion map $X(0) \hookrightarrow Y(0) \hookrightarrow Y(1) \hookrightarrow \cdots Y(m+1) = \Delta ^ m \times \Delta ^2$ is inner anodyne, as desired. $\square$