The infinity-connected factorization system

(1) The \(n\)-connected maps form a saturated class (i.e. closed under pushouts and closed under colimits in \(\Ar(T)\)), by the factorization system from Proposition 3.18. In particular, the \(\infty\)-connected maps form a saturated class.For the 2-out-of-3 property, it remains to check left cancellation. If \(gf\) and \(g\) are \(\infty\)-connected, then for every \(n\) the map \(gf\) is \(n\)-connected and \(g\) is \((n+1)\)-connected, so Corollary 3.33 shows that \(f\) is \(n\)-connected. Thus \(f\) is \(\infty\)-connected. Closure under base change follows from part (2) of Proposition 3.18. Closure under diagonals follows from Theorem 3.22: if \(f\) is \(\infty\)-connected, then for every \(n\) it is \((n+1)\)-connected, and hence \(\Delta_f\) is \(n\)-connected.Small generation is the additional accessibility input. The full subcategory of \(\Ar(T)\) spanned by the \(\infty\)-connected morphisms is accessible, and therefore the resulting strongly saturated class is of small generation. We refer to [Lurie 2009, Proposition 6.5.2.8] for this argument.(2) The \(\infty\)-truncated objects are precisely the local objects with respect to this strongly saturated class of small generation. The existence and accessibility of the reflector \(\tau_{\infty}\) therefore follow from Proposition A.15, or equivalently from [Lurie 2009, Proposition 5.5.4.15].For (3), it remains to show that any morphism \(f\colon X \to Y\) can be factored into an \(\infty\)-connected morphism followed by an \(\infty\)-truncated one. The same argument as in part (1) of Proposition 3.18 shows that the map \(X \to \tau_{\infty}(X/Y)\) provides such a factorization.