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What are concrete and abstract examples of problems (even whole programmes of inquiry) when one has a choice to use a "derived" theory (e.g., $\infty$-categories, DAG, HAG, $DRep_k(G)$, "higher" counting,...) or its non-derived variant (1-categorical arguments, basic algebraic geometry, "stupid" counting, $Rep_k(G)$ ...) but the derived variant will not work?

Motivation: when defining, say, some geometric theory, is it still useful to write up a 1-categorical treatment distinct from an $\infty$-categorical treatment?

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    $\begingroup$ Where is the Kool-Aid? $\endgroup$
    – nfdc23
    Commented Apr 14, 2017 at 15:17
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    $\begingroup$ In the paper arXiv:1610.07421v4 "Modelling and Computing Homotopy Types: I" I explain how the use of certain strict higher homotopy groupoids has led to a new approach to algebraic topology at the border between homology and homotopy and to new computational procedures. Work on (weak) $\infty$-groupoids has not to my knowledge yielded corresponding results. $\endgroup$ Commented Apr 14, 2017 at 15:44
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    $\begingroup$ The first paragraph is fine, but the "Motivation" (which was what I first read) is surreal; e.g., EGA is "1-categorical" and has vast depth! Not only is it good to write up explanations of things in terms that more people can understand, but if you think about how to explain ideas in as concrete a form as possible then you will come to a better understanding of where the real substance lies underneath a lot of abstract nonsense (e.g., articulate exactly why certain heavier machines are serving a real purpose in a given setting, if so). Talk with algebraic geometers in your department. $\endgroup$
    – nfdc23
    Commented Apr 14, 2017 at 17:54
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    $\begingroup$ It is of course fairly common that the passage from the strict category to the derived category is a loss of information. I find it interesting that QCoh(P^1) is very far from being a category of modules for a unital associative ring. The derived category is famously the modules of a quiver algebra. $\endgroup$ Commented Apr 15, 2017 at 2:48
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    $\begingroup$ Please speak with some experienced algebraic geometry faculty in your department; nearly all of the material in algebraic geometry books and journal articles has nothing whatsoever to do with $\infty$-categorical methods (and Artin proved deep theorems on Artin stacks perfectly well without such things). It doesn't make sense why you should feel the need to keep explaining why you aren't doing things $\infty$-categorically; do whatever is appropriate to a problem at hand, and ignore any pressure to drink the Kool Aid if you can make do with water (which is better for your health too). $\endgroup$
    – nfdc23
    Commented Apr 15, 2017 at 4:11

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The comments are long, so I will post this as an answer. One example that has both a classical and a derived aspect is the construction, in algebraic geometry, of the virtual fundamental class by Kai Behrend and Barbara Fantechi. In Kontsevich's paper, "Enumerating Rational Curves via Torus Actions", he proposed a derived approach. However, there are several difficult things to check to even make sense of the construction, and there are many other compatibilities in order to prove that the construction satisfies the Kontsevich-Manin axioms for Gromov-Witten invariants. These things need to be proved with "classical" arguments about cycles on algebraic varieties and cotangent complexes. What I find most beautiful about the paper of Behrend-Fantechi is that they also clarify longstanding constructions like the excess intersection formula with their approach.

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    $\begingroup$ Perfect. Precisely the type of situation I wanted to hear about. Thank you so much Jason! $\endgroup$ Commented Apr 15, 2017 at 20:53

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