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Lurie's book Higher Topos Theory is extremely interesting, but pretty overwhelming. I don't have the time to read it at the moment. However, the last chapter (7) gives applications of $\infty$-topoi in topology and I would like to know rough things about these applications - maybe somebody of you wants to give a summary. :-)

  • In which way are $\infty$-topoi useful in topology? Just as a tool to state a theorem (elegantly) or as a tool to prove a theorem?
  • If $\infty$-topoi are used to prove theorems, which theorems about $\infty$-topoi are used? (I am curious because I don't even know a proper application of 1-topoi to topology.)
  • The preface talks about how $\infty$-topoi solve the problem of understanding $H^n(X;G)$ for $n>1$ conceptually, similar to how $H^1(X;G)$ can be understood conceptually as being represented by the the classifying space $BG$, which classifies $G$-torsors. How do $\infty$-topoi help understanding $H^n(X;G)$ for $n>1$? How is that related to stacks? (I guess the preface is trying to explain exactly that - but I just don't understand what he is trying to say starting at page viii.)

What I find surprising: $\infty$-topoi seem to be useful for completely different things than 1-topoi. 1-topoi were invented to invent étale cohomology. But I don't see any $\infty$-étale cohomology in Lurie's book.

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    $\begingroup$ étale cohomology isn't much useful in topology - more in algebraic geometry. So a list of applications in topology wouldn't necessarily include generalizations of étale cohomology. $\endgroup$
    – Will Sawin
    Commented Feb 6, 2022 at 21:07
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    $\begingroup$ It's probably more in shape theory, which deals with the algebraic topology of spaces for which the usual invariants like homotopy groups and singular cohomology fail to detect nontrivial topology. If a topological space is so bad that there are no non-constant maps from standard simplices, then singular cohomology is trivial. Similarly if the topology is "too global" to detect with maps from boundaries of simplices (eg the Warsaw circle). $\endgroup$
    – David Roberts ♦
    Commented Feb 7, 2022 at 0:49

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