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 ∞-categories $\infty$-categories for years, and the idea of (∞,n)-categories $(\infty,n)$-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.?
EDIT: To clarify the question: An example of a key insight would be the use of (∞,n)-categories with problems of type X. But what is it exactly about Jacob's construction of (∞,n)-categories which is superior to previous constructions- can somebody pinpoint the conceptual breakthough?It's more than just "abstracter than thou" (which I'm not sure is even true in all cases ).
