Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Variant 4.3.6.18. Let $p$ and $q$ be nonnegative integers. Then the pushout-join monomorphism

\[ (\operatorname{\partial \Delta }^ p \star \Delta ^ q) \coprod _{ ( \operatorname{\partial \Delta }^ p \star \operatorname{\partial \Delta }^ q) } ( \Delta ^ p \star \operatorname{\partial \Delta }^ q) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]

of Construction 4.3.6.3 is isomorphic to the boundary inclusion $\operatorname{\partial \Delta }^{p+1+q} \hookrightarrow \Delta ^{p+1+q}$.

Proof. We proceed as in the proof of Lemma 4.3.6.16. Let $u: \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}$ be the isomorphism supplied by Example 4.3.3.14, and let $\sigma $ be an $n$-simplex of the join $\Delta ^{p} \star \Delta ^{q}$. We wish to show that $u(\sigma )$ belongs to the boundary $\operatorname{\partial \Delta }^{p+1+q}$ if and only if $\sigma $ belongs to the union of the simplicial subsets

\[ \operatorname{\partial \Delta }^ p \star \Delta ^{q} \subseteq \Delta ^{p} \star \Delta ^{q} \supseteq \Delta ^{p} \star \operatorname{\partial \Delta }^{q}. \]

We consider three cases (see Remark 4.3.3.17):

  • The simplex $\sigma $ belongs to the simplicial subset $\Delta ^{p} \subseteq \Delta ^{p} \star \Delta ^{q}$. In this case, $\sigma $ is contained in $\Delta ^{p} \star \operatorname{\partial \Delta }^{q}$ and $u(\sigma )$ is contained in $\operatorname{\partial \Delta }^{p+1+q}$.

  • The simplex $\sigma $ belongs to the simplicial subset $\Delta ^{q} \subseteq \Delta ^{p} \star \Delta ^{q}$. In this case, $\sigma $ is contained in $\operatorname{\partial \Delta }^{p} \star \Delta ^{q}$ and $u(\sigma )$ is contained in $\operatorname{\partial \Delta }^{p+1+q}$.

  • The simplex $\sigma $ factors as a composition

    \[ \Delta ^{n} = \Delta ^{p' + 1 + q'} \simeq \Delta ^{p'} \star \Delta ^{q'} \xrightarrow { \sigma _{-} \star \sigma _{+} } \Delta ^{p} \star \Delta {q}. \]

    In this case, $\sigma $ belongs to the union $(\operatorname{\partial \Delta }^{p} \star \Delta ^{q}) \cup ( \Delta ^{p} \star \operatorname{\partial \Delta }^{q} )$ if and only if either $\sigma _{-}$ or $\sigma _{+}$ fails to be surjective at the level of vertices. This is equivalent to the requirement that the map $u(\sigma ): \Delta ^{n} \rightarrow \Delta ^{p+1+q}$ fails to be surjective at the level of vertices: that is, it is a simplex of the boundary $\operatorname{\partial \Delta }^{p+1+q}$.

$\square$