TITLE: To derive or not to derive, that is the question
QUESTION [5 upvotes]: 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?

REPLY [8 votes]: 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.