Higher topos and derived algebraic geometry are relatively new areas and probably fewer people are working on them compared to the majority of topologists or geometers. I believe higher categories should have great power in producing new results in algebra (K-theory, algebraic topology, algebraic geometry). What are some important applications of higher categories in these areas?

The use of higher categories seems to be closer to the nature of homotopy theory and algebraic K-theory. I have heard from people in algebraic K-theory that higher categories are very useful and can produce important new results. My feeling is that applications of higher categories in AG are less substantial as most algebraic geometers don't need many abstract settings. I will appreciate suggestions for references or overviews of applications in higher categories in K-theory, algebraic topology, algebraic geometry or homotopy theory.

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    $\begingroup$ 'Must one obtain a very deep knowledge of AG or AT to understand the applications of higher topos properly?' I doubt that, since the applications seem to be of a more formal nature. I think most working algebraic geometers don't know what a topos is (even Serre admitted that he doesn't know), and I think this says much regarding the importance of topos theory in algebraic geometry. $\endgroup$ – Bernie Feb 26 at 23:28
  • $\begingroup$ I heard higher topos theory and derived algebraic geometry are crucial in String field theory, a branch of mathematical physics. $\endgroup$ – Andrews Feb 29 at 6:53
  • $\begingroup$ In String field theory, I've seen little or know typos theory. Drived' stuff is very useful but need not be in AG. See Higher Structures' If it doesn't endanger you, where are you currently $\endgroup$ – Jim Stasheff Mar 3 at 20:22
  • $\begingroup$ Yes, the journal $\endgroup$ – Jim Stasheff Mar 4 at 22:13
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    $\begingroup$ This paper arxiv.org/pdf/1102.1150.pdf shows how the derived geometry (or homotopical) approach allows to get functoriality "for free" in the construction of perfect obstruction theories. $\endgroup$ – DamienC Mar 8 at 23:45

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