# Why do we care about $(\infty,2)$-categories?

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?

• A related/relevant MO question: mathoverflow.net/questions/373079 Aug 10, 2021 at 21:25
• There are lots of [$\infty$-] categories we might care about in mainstream math, and performing operations on them often requires a formalism of [$\infty$-]2-categories.. One example (not unrelated to my favorite, the TFT one) is treating the functoriality of categories of sheaves under correspondences - this is the main point of the book of Gaitsgory-Rozenblyum. Aug 11, 2021 at 2:32
• The three examples - 2-categories of bordisms, 2-categories of spaces with correspondences, and 2-categories of categories -- come packaged together when we study Lagrangian topological field theories (in codim 2) which are functors from the first to the third that factor through the second. Aug 11, 2021 at 2:34
• One more example (again not at all unrelated): monoidal categories come up in many contexts in math. Their derived version is most naturally studied from the point of view of $(\infty,2)$-categories (the Morita theory of monoidal categories forms such a beast, just like the Morita theory of algebras is one of the most natural sources of ordinary categories) Aug 11, 2021 at 2:36
• I have a kind of déjà-vu. 15 years ago we were challenged to explain why to step from 1-categories to 2-categories. Now we have progressed to an $\infty$-analogue :-) See mathoverflow.net/questions/82981/are-higher-categories-useful/… Aug 11, 2021 at 7:44

As you say, a major use of $$(\infty,2)$$-categories is for organising $$(\infty,1)$$-categories and similar objects (stable $$\infty$$-categories, $$\infty$$-topoi, enriched $$\infty$$-categories, $$\infty$$-operads...). The importance of the non-invertible 2-cells is the same as in classical $$(2,2)$$-category theory: they provide natural internal notions of adjunction, base change mappings, lax functor, lax monoidal functor, Kan extension, and so on. An $$(\infty,2)$$-category can be used to organise collections of these structures and keep track of coherences between them.

A good illustration of the utility of this is the notion of a "six-functor formalism," which Gaitsgory and Rozenblyum (https://bookstore.ams.org/surv-221/) argued is best captured by a certain symmetric monoidal $$(\infty,2)$$-functor on an $$(\infty,2)$$-category of correspondences between derived stacks. We definitely need $$\infty$$ here because the value of such a functor would be something like the derived $$\infty$$-category of quasi-coherent or constructible sheaves, and the source may also include some derived/higher objects. We definitely need $$2$$ because the 2-cells of the category of correspondences encode all kinds of coherences between the six functors (for example, base change 2-cells and the higher associativity of compositions of 2-d grids of base change squares). Even if one ultimately only cares about constructing functors out of correspondences on a 1-category, in practice one still needs its universal property among $$(\infty,2)$$-categories.

See also my paper https://arxiv.org/abs/2005.10496 for a slightly different take to Gaitsgory-Rozenblyum's.

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.

• “local geometric Langlands is often stated as an equivalence of (∞,2)-categories”: what are some (good) references in which LGL is often stated in this form? Aug 14, 2021 at 19:24
• @DmitriPavlov See for instance Section 1.2 of arxiv.org/abs/1310.5127v2. The original philosophical source is arxiv.org/abs/math/0508382, but that was written at a time when the foundations were insufficient to make such a statement rigorous.
– dhy
Aug 14, 2021 at 19:45
• @DmitriPavlov That being said, the unfortunate reality at the moment is that the published literature lags behind "folklore."
– dhy
Aug 14, 2021 at 19:50