The content of my previous ''Fun with gerbes" talk (of which the current talk is wholly independent) was etale homotopy 2-types via Grothendieck's theory of champs (the translation stack is awful), and the content of this talk will be applications to Lefshetz theorems. A particular feature of having the right definition of higher homotopy groups via n-categories is that the Lefschetz theorem becomes a more or less tautological induction, and at the very least I'll explain the initial $\pi_0$ step in the induction- typically all problems in this area are, when correctly understood, are problems about $\pi_0$. A manuscript containingi the material from the "Fun with gerbes" series is available here: http://arxiv.org/abs/1507.00797