Lemma 5.17.
Let \(u\) and \(v\) be maps in \(T_{/Z}\). If \(u\) is \(m\)-connected and \(v\) is \(n\)-connected, then \(u \square_Z v\) is \((m+n+2)\)-connected.
Proof
We work in the topos \(T_{/Z}\). Choose a small collection \(\Gg\) of generators. The class of \(m\)-connected maps is the saturation of the maps
\[S^{m+1}\otimes U\longrightarrow U,
\qquad U\in\Gg.\]
Indeed, a map is right orthogonal to these generators precisely when its fibers are \(m\)-truncated, by the characterization of truncated objects in Section 3.1. The same description holds for the \(n\)-connected maps.The pushout product preserves colimits separately in both variables. It therefore suffices to take \(u\) and \(v\) of the displayed form, allowing arbitrary objects \(U\) and \(V\). Their pushout product is \[(S^{m+1}*S^{n+1})\otimes(U\times_ZV)
\longrightarrow U\times_ZV,\]
where \(*\) denotes the join of animae. Since \(S^{m+1}*S^{n+1}\iso S^{m+n+3}\), this map is \((m+n+2)\)-connected. The \((m+n+2)\)-connected maps form a saturated class, so the same follows for arbitrary \(u\) and \(v\).