TITLE: Why do we care about $(\infty,2)$-categories?
QUESTION [13 upvotes]: crossposted from MSE as suggested by Igor Sikora
Homotopy theory provides much motivation for studying $(\infty,1)$-categories in their relations to homotopical algebra, derived geometry, stable homotopy stuffs, cohomology, physics, and so on. As for $2$-categories, one doesn't even have to motivate them since they're all over the place.
However, I'm having a hard time motivating myself to study $(\infty,2)$-categories. I've been learning a bunch of facts about them: how the Duskin nerve can be regarded as an embedding from bicategories to the complicial sets model, how the Lack-Paoli nerve can be regarded as an embedding to a "simplicially enriched model", but in the end I can't see why we would want to deal with $(\infty,2)$-categories in the first place.
All I've seen so far is their use in low dimensional TQFT, and also as a way to encode the $(\infty,2)$-category of $(\infty,1)$-categories, both in the context of specific models as well as in $\infty$-cosmological contexts.
So (do we care, and if so) why do we care about $(\infty,2)$-categories?

REPLY [10 votes]: One place where $(\infty,2)$-categories shows up is the geometric Langlands program. (As in David Ben-Zvi's comment, this is again related to the TFT example.) Indeed, local geometric Langlands is often stated as an equivalence of $(\infty,2)$-categories:
$$F:D(G((t)))\operatorname{-mod}\cong\operatorname{QCoh}(\operatorname{LocSys}_{\check{G}}(\overset{\circ}{D}))\operatorname{-mod}.$$
(Actually, this is false as stated - the RHS is modified in the actual conjecture, but it's a little hard to state this modification.) Here $G((t))$ is the loop group, and $\operatorname{LocSys}_{\check{G}}(\overset{\circ}{D})$ is the moduli space of $\check{G}$-local systems on the punctured disk. $D(X)$ and $\operatorname{QCoh}(X)$ denote the derived ($\infty-$) categories of D-modules and quasicoherent sheaves on $X$, respectively. The $(\infty,2)$-categories in question are categories of modules over the respective monoidal $(\infty,1)$-categories.
By the way, an $(\infty,2)$-category is an enormous amount of data, and so an equivalence of $(\infty,2)$-categories is an extremely powerful statement. For any objects $X,Y\in D(G((t)))\operatorname{-mod},$ the above conjecture predicts an equivalence (of ($\infty,1)$-categories) $\operatorname{Hom}(X,Y)\cong\operatorname{Hom}(F(X),F(Y)).$
At this point we know what $F(X)$ should be for quite a few $X,$ which makes it possible to extract a lot of nontrivial equivalences from local geometric Langlands. The simplest example is to take $X=Y=D(\operatorname{Gr}),$ $\operatorname{Gr}$ the affine Grassmannian. In this case $\operatorname{End}_{G((t))}(D(\operatorname{Gr}))$ can be identified with the derived Satake category, i.e., the derived category of $G[[t]]$-equivariant D-modules on $\operatorname{Gr}$, and you recover the derived geometric Satake equivalence.