Corollary 3.40.
The left adjoint \(\tau_{\infty}\colon T \to T_{\leq \infty}\) is left exact. In particular, \(T_{\leq \infty}\) is again a topos.
Proof
We first explain why the \(\infty\)-connected maps are closed under finite limits in \(\Ar(T)\). They contain the terminal object. For pullbacks, consider a morphism between cospans whose three component maps are \(\infty\)-connected. The induced map between their pullbacks factors as
\[X\times_ZY \longrightarrow X\times_{Z'}Y \longrightarrow X'\times_{Z'}Y \longrightarrow X'\times_{Z'}Y'.\]
The last two maps are base changes of \(X\to X'\) and \(Y\to Y'\), while the first is a base change of the diagonal of \(Z\to Z'\). Each factor is therefore \(\infty\)-connected by Proposition 3.39, and so is their composite. This is the same three-factor argument used in the proof of Lemma 4.15.For a finite diagram \(\{X_i\}_{i \in I}\) in \(T\), it follows that the map \(\lim_i X_i \to \lim_i \tau_{\infty}X_i\) is \(\infty\)-connected. Since \(\infty\)-truncated objects are local objects for a class of morphisms, they are closed under limits, so the target is \(\infty\)-truncated. The map therefore exhibits the target as the hypercompletion of \(\lim_iX_i\), giving an isomorphism \[\tau_{\infty}(\lim_i X_i) \iso \lim_i \tau_{\infty}X_i.\]