How do $\infty$-categories allow us to do descent on the derived level?

Question

I have heard that one application of $\infty$-categories is that they allow us to formulate a meaningful theory of descent for derived categories (say of sheaves on a scheme). While I'm sure the details are somewhere in Lurie's exposition of stable $\infty$-categories, I was hoping that someone familiar with the process could explain in broad strokes why we can't do this in the classical setting, and what $\infty$-categories add to the picture that changes the situation.

Answer 1

Let $X$ be a topological space covered by open sets $U$ and $V$. Let $\mathscr{F}$ and $\mathscr{G}$ be complexes of sheaves defined on $U$ and $V$, respectively. Suppose you are given an isomorphism $\alpha: \mathscr{F}|_{ U \cap V} \rightarrow \mathscr{G}|_{ U \cap V}$ in the derived category of the intersection $U \cap V$. You would like to use these to glue $\mathscr{F}$ and $\mathscr{G}$ together to obtain a complex of sheaves on $X$. Let $j: U \hookrightarrow X$, $j': V \hookrightarrow X$, and $j'': U \cap V \hookrightarrow X$ denote the inclusion maps. Then the "glued" complex should be the fiber of the map $$j_{\ast} \mathscr{F} \oplus j'_{\ast} \mathscr{G} \rightarrow j''_{\ast} \mathscr{G}|_{U \cap V},$$ which is given on the first factor by $\alpha$. Working at the level of triangulated categories, this characterizes the glued complex up to non-canonical isomorphism. But for many purposes, producing a complex which is only well-defined up to non-canonical isomorphism is probably not good enough: you would like to define something that depends functorially on the input. The formalism of triangulated categories is poorly suited to this, because taking the fiber (or cocone) of a morphism is not a functorial operation. This is the sort of thing that is "corrected" by working with $\infty$-categories.