Lemma 4.8.8.18 (06Y4)

Lemma 4.8.8.18. Let $m$ and $n$ be nonnnegative integers. Then the inclusion map

\[ \iota _{m,n}: ( \Delta ^{m} \times \operatorname{\partial \Delta }^{n} ) \coprod _{ ( \operatorname{\partial \Delta }^{m} \times \operatorname{\partial \Delta }^{n} ) } ( \operatorname{\partial \Delta }^{m} \times \Delta ^{n} )\hookrightarrow \Delta ^{m} \times \Delta ^{n} \]

is categorically $(m+n-1)$-connective.

Proof. Without loss of generality, we may assume that $m \leq n$.If $m = 0$, then $\iota _{m,n}$ can be identified with the inclusion map $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^ n$, which is categorically $(n-1)$-connective by virtue of Example 4.8.8.16. We may therefore assume that $m \geq 1$, so that $n \geq 1$. We proceed by induction on $n$. If $n > 1$, then we can choose an integer $0 < i < n$, so that $\operatorname{\partial \Delta }^{n}$ contains an inner horn $\Lambda ^{n}_{i}$. In this case, Lemma 1.5.7.5 guarantees that the composite map

\[ ( \Delta ^{m} \times \Lambda ^{n}_{i} ) \coprod _{ ( \operatorname{\partial \Delta }^{m} \times \Lambda ^{n}_{i} ) } ( \operatorname{\partial \Delta }^{m} \times \Delta ^{n} ) \xrightarrow {\rho } ( \Delta ^{m} \times \operatorname{\partial \Delta }^{n} ) \coprod _{ ( \operatorname{\partial \Delta }^{m} \times \operatorname{\partial \Delta }^{n} ) } ( \operatorname{\partial \Delta }^{m} \times \Delta ^{n} ) \xrightarrow { \iota _{m,n} } \Delta ^{m} \times \Delta ^{n} \]

is inner anodyne, and therefore categorically $(m+n-1)$-connective. Consequently, to prove that $\iota _{m,n}$ is categorically $(m+n-1)$-connective, it will suffice to show that $\rho $ is categorically $(m+n-2)$-connective (Remark 4.8.8.6). This follows from Proposition 4.8.8.12, since $\rho $ is a pushout of $\iota _{m,n-1}$ (which is categorically $(m+n-2)$-connective by our inductive hypothesis).

It remains to treat the case $m = n = 1$. Let $K$ be a simplicial subset of $\Delta ^1 \times \Delta ^1$ obtained by removing a single nondegenerate $2$-simplex. In this case, we observe that $\iota _{m,n}$ factors as a composition

\[ ( \Delta ^{1} \times \operatorname{\partial \Delta }^{1} ) \coprod _{ ( \operatorname{\partial \Delta }^{1} \times \operatorname{\partial \Delta }^{1} ) } ( \operatorname{\partial \Delta }^{1} \times \Delta ^{1} ) \hookrightarrow K \xrightarrow {\iota '} \Delta ^1 \times \Delta ^1, \]

where the map on the left is inner anodyne (it is a pushout of the inner horn inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$). We are therefore reduced to proving that $\iota '$ is categorically $1$-connective, which is a special case of Example 4.8.8.16 (since $K$ contains the $1$-skeleton of $\Delta ^1 \times \Delta ^1$). $\square$