9
$\begingroup$

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.

$\endgroup$
1

1 Answer 1

23
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.