Kerodon

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

Proposition 1.1.3.11. Let $S^{-}_{\bullet }$ and $S^{+}_{\bullet }$ be simplicial sets having dimensions $\leq k_{-}$ and $\leq k_{+}$, respectively. Then the product $S^{-}_{\bullet } \times S^{+}_{\bullet }$ has dimension $\leq k_{-} + k_{+}$.

Proof. Let $\sigma = (\sigma _{-}, \sigma _{+})$ be a nondegenerate $n$-simplex of the product $S^{-}_{\bullet } \times S^{+}_{\bullet }$. Using Proposition 1.1.3.4, we see that $\sigma _{-}$ and $\sigma _{+}$ admit factorizations

\[ \Delta ^{n} \xrightarrow { \alpha _{-} } \Delta ^{n_{-}} \xrightarrow { \tau _{-} } S^{-}_{\bullet } \quad \quad \Delta ^{n} \xrightarrow { \alpha _{+} } \Delta ^{n_{+}} \xrightarrow { \tau _{+} } S^{+}_{\bullet }, \]

where $\tau _{-}$ and $\tau _{+}$ are nondegenerate, so that $n_{-} \leq k_{-}$ and $n_{+} \leq k_{+}$. It follows that $\sigma $ factors as a composition

\[ \Delta ^{n} \xrightarrow { (\alpha _{-}, \alpha _{+} ) } \Delta ^{n_{-}} \times \Delta ^{n_{+}} \xrightarrow { \tau _{-} \times \tau _{+} } S_{\bullet }^{-} \times S_{\bullet }^{+}. \]

The nondegeneracy of $\sigma $ guarantees that the map of partially ordered sets $[n] \xrightarrow { (\alpha _{-}, \alpha _{+} )} [n_{-} ] \times [n_{+} ]$ is a monomorphism, so that $n \leq n_{-} + n_{+} \leq k_{-} + k_{+}$. $\square$