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I think, one of the key insights underlying derived algebraic geometry and Lurie's treatment of elliptic cohomology is taking some ideas of Grothendieck really serious. Two manifestations:

1) One of the points of the scheme theory initiated by Grothendieck is the following: if one takes intersection of two varieties just on the point-set level, one loses information. One has to add the possibility of nilpotents (somewhat higher information) to preserve the information of intersection multiplicities and get the "right" notion of a fiber product. Now one of the points of derived algebraic geometry (as explained very lucidly in the introduction to DAG V) is that for homological purposes this is not really the right fiber product - you need to take some kind of homotopy fiber product. This is, because one still loses information because one is taking quotients - one should add isomorphism instead and view it on a categorical level. Thus, you can take a meaningful intersection of a point with itself, for example. This is perhaps an instance where the homological revolution, which went to pure mathematics last century, benefits from a second wave, a homotopical revolution - if I am allowed to overstate this a bit.

2) Another insight of Grothendieck and his school was, how important it is to represent functors in algebraic geometry - regardless of what you want at the end. [as Mazur reports, Hendrik Lenstra was once sure that he did want to solve Diophantine equations and did not want to represent functors - and later he was amused that he represented functors to solve Diophantine equations.] And this is Lurie's approach to elliptic cohomology and tmf: Hopkins and Miller showed the existence of a certain sheaf of $E_\infty$-ring spectra on the moduli stack of elliptic curves. Lurie showed that this represents a derived moduli problem (of oriented derived elliptic curves).

Also his solution of the cobordism hypothesis has a certain flavor of Grothendieck: you have to put things in a quite general framework to see the essence. This philosophy also shines quite clearly through his DAG, I think.

Besides, I do not think, there is a single key insight in Higher Topos Theory besides the feeling that infinity-categories are important and that you can find analogues to most of classical category theory in quasi-categories. Then there are lots of little (but every single one amazing) insights, how this transformation from classical to infinity-category theory works.

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I think, one of the key insights underlying derived algebraic geometry and Lurie's treatment of elliptic cohomology is taking some ideas of Grothendieck really serious. Two manifestations:

1) One of the points of the scheme theory initiated by Grothendieck is the following: if one takes intersection of two varieties just on the point-set level, one loses information. One has to add the possibility of nilpotents (somewhat higher information) to preserve the information of intersection multiplicities and get the "right" notion of a fiber product. Now one of the points of derived algebraic geometry (as explained very lucidly in the introduction to DAG V) is that for homological purposes this is not really the right fiber product - you need to take some kind of homotopy fiber product. This is, because one still loses information because one is taking quotients - one should add isomorphism instead and view it on a categorical level. Thus, you can take a meaningful intersection of a point with itself, for example. This is perhaps an instance where the homological revolution, which went to pure mathematics last century, benefits from a second wave, a homotopical revolution - if I am allowed to overstate this a bit.

2) Another insight of Grothendieck and his school was, how important it is to represent functors in algebraic geometry - regardless of what you want at the end. [as Mazur reports, Hendrik Lenstra was once sure that he did want to solve Diophantine equations and did not want to represent functors - and later he was amused that he represented functors to solve Diophantine equations.] And this is Lurie's approach to elliptic cohomology and tmf: Hopkins and Miller showed the existence of a certain sheaf of $E_\infty$-ring spectra on the moduli stack of elliptic curves. Lurie showed that this represents a derived moduli problem (of oriented derived elliptic curves).

Also his solution of the cobordism hypothesis has a certain flavor of Grothendieck: you have to put things in a quite general framework to see the essence. This philosophy also shines quite clearly through his DAG, I think.