TITLE: What are Jacob Lurie's key insights?
QUESTION [68 upvotes]: This question is inspired by this Tim Gowers blogpost.
I have some familiarity with the work of Jacob Lurie, which contains ideas which are simply astounding; but what I don't understand is which key insight allowed him to begin his programme and achieve things which nobody had been able to achieve before. People had looked at $\infty$-categories for years, and the idea of $(\infty,n)$-categories is not in itself new. What was the key new idea which started "Higher Topos Theory", the proof of the Baez-Dolan cobordism hypothesis, "Derived Algebraic Geometry", etc.?

REPLY [21 votes]: I think one of the insights leading to his successes, which is of course not unique to him, is that when doing higher category theory it is useful to add invertible higher cells going all the way up to the top before you try to add noninvertible ones anywhere.  This is along the same lines as Noah's answer: the cobordism hypothesis, which was originally stated in terms of n-categories, becomes easier once you generalize it to a stronger statement about (∞,n)-categories, since then induction becomes easier/possible.