Lemma 6.49. ([Lurie 2009, Lemma 6.5.3.10])
Let \(f_\bullet\colon U_\bullet \to V_\bullet\) be a morphism of simplicial objects in \(T\). Suppose that \(f_k\colon U_k \to V_k\) is an isomorphism for all \(k \leq n\). Then the induced map \(\abs{f_\bullet}\colon \abs{U_\bullet} \to \abs{V_\bullet}\) is \(n\)-connected.
Proof
This is [Lurie 2009, Lemma 6.5.3.10]. Lurie reduces first to animae and then proves the claim using projectively cofibrant simplicial Kan-complex models.
References
- Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.