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?

stricthigher 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. $\endgroup$exactly whycertain heavier machines are serving a real purpose in a given setting, if so). Talk with algebraic geometers in your department. $\endgroup$interestingthat 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$14more comments