Why higher category theory?

Question

This is a soft question. I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I have some background in commutative algebra and algebraic topology and I am learning AG quite intensively. I understand that category theory is useful for generalizing different mathetical objects, and in the field of AT, it makes the idea of homology and homotopy so clear as they are just functors from (based) space to the category of (abelian) groups, and many things are just limits or colimts; I also do understand you need the notion of abelian categories for homological algebra and constructing sheaves like the grand project done by Grothendieck.

However, I still don't understand why study such elusive things like higher category theory where many things defy intuition? I have talked to several top mathematicians in my department, many of whom apperently think it is useless. However, it should not be as celebrated mathematicians like Jacob Lurie is making a lot of effort trying to understand it. So can anyone explain to me why study higher category theory? What problems are they most concerned with in higher category theory and what connections are there with AT or AG?

Thank you very much in advance.

Answer 1

This is addressed to Paul Siegel's answer, which I find misleading, and not just because it is not true that maps that induce the same map on homotopy groups are homotopic. (The famous and open Freyd conjecture asserts that maps of finite spectra that induce the same map on homotopy groups are homotopic, but that is digressive).

It is the comments about stable homotopy theory that I find most misleading. Good modern categories of spectra are symmetric monoidal under the smash product, and in such categories $E_{\infty}$ ring spectra as first defined in 1972 are equivalent to commutative monoids (as defined in any symmetric monoidal category). The higher homotopies are then hidden in the definition of the smash product. For purposes of calculation, this is extremely convenient and useful, and there is little point to invoking higher category theory of any sort. For example, people doing serious computations with TMF don't invoke higher category theory.

Moreover, higher category theory should not be confused with just the theory of $(\infty,1)$-categories. Those are central to categorical homotopy theory, but higher category theory has many other incarnations. For example, $2$-category theory as developed in Australia is hugely rich and powerful in applications. In the hands of Riehl and Verity it has very important applications even to the theory of $(\infty,1)$ categories, but it is very concretely useful in a slew of other areas, including stable homotopy theory. Moreover, there are quite distinct and quite different theories of $n$-categories, which start with sets and categories at levels $0$ and $1$, and $(\infty,n)$-categories, which can start with simplicial sets and simplicially enriched categories at levels $0$ and $1$ (there are other possible choices).

Both 2-category theory and $(\infty,1)$-category are examples where there are lots of applications outside the theory itself, just as is true of ordinary category theory.

In addition to the sources others have cited, there is a book ``Towards higher categories'' with articles that discuss various aspects of higher category theory and that may give some idea of the why as well as the wherefore.

Answer 2

Here are some reasons. I also would like to emphasize the distinction between $(\infty,1)$-categorical thinking, and $(\infty,n)$-categorical thinking, so I'll give examples of both. These are non-exhaustive, non-universal reasons, and merely examples of where higher category has been useful. I'll also try to avoid talking about reasons that other people on this thread have already mentioned.

In the $(\infty,1)$-setting:

• Mapping cones are not functorial for triangulated categories, while in the dg setting, or the stable oo-category setting, they are. This makes many statements in homological algebra more natural. This functoriality is not functoriality in the sense of usual 1-category theory, but it is functoriality for a setting in which one acknowledges that there is a space of morphisms, and a notion of a homotopy colimit. So here, passing to higher category theory gives you a nice algebraic fix.

• While sometimes you can hide away homotopical issues into strict, non-homotopical guises (the way you can have Eoo ring spectra become honest commutative objects), sometimes you don't want to have to build the machinery to strictify some wrinkles in the carpet while pushing wrinkles elsewhere. Especially when talking about functorial constructions and (co)limits, many diagrams come to you naturally begging for recognition of higher-homotopical structures.

• Statements in symplectic geometry relating geometric invariants to algebraic structures cannot be proven without taking into account higher-categorical structures. For instance, in proving that quantum cohomology is isomorphic to Hochschild cohomology of the Fukaya category, one can't just remember the cohomology groups of the Fukaya category. We need the full knowledge of the Aoo operations, and the Aoo operations encode homotopies and higher homotopies.

In the $(\infty,n)$-setting:

• The category of algebraic varieties naturally looks like an $(\infty,2)$-category. The objects are varieties. Between any two of them, one then has the dg-category of sheaves on $X \times Y$. The $2$-morphisms are morphisms in this category. This, of course, has a natural functor to the $2$-category of stable categories, bimodules, and maps between bimodules. This is a global way to encode what algebraic properties one sees out of Fourier-Mukai transform of sheaves.
• On the mirror side, the category of symplectic manifolds forms an $(\infty,2)$-category. The objects are certain symplectic manifolds with decorations, and between any two of them, one has the Fukaya category of the product (with orientation reversed on one of the factors). This $(\infty,2)$-category still only exists conjecturally at this point, but is expected to spell out the algebraic properties of the geometric notion of how Floer cohomology plays with composition of Lagrangian correspondences. It also offers a conceptual explanation of why one would expect quantum cohomology to be equivalent to Hochschild cohomology of a Fukaya category.
• In derived algebraic geometry, one sees a notion of shifted symplectic manifolds and Lagrangians inside them. This is expected to naturally fit together into an $(\infty,n)$-category where objects at different levels are given by shifted symplectic manifolds arising as derived intersections of Lagrangians. We expect this to encode the higher structures present in field theories with boundary conditions, and this is a very exciting avenue of research at present.
• Also, the cobordism hypothesis (which requires higher category theory to state) proves that the geometric content of manifolds and cobordisms between them captures a very algebraic condition of having duals and higher adjoints. This can be viewed as a demonstration that higher categorical structures can reflect the geometric data of manifolds.

In thinking about monoidal structures:

• Secretly, when you understand that natural transformations between functors are important, you have already recognized the $2$-categorical structure in the category of categories. So higher categories are not useless here, for sure. For instance, do you ever use symmetric monoidal categories?
• Modular tensor categories, braided monoidal categories, etc, are used in defining the structural requirements needed for defining invariants of $3$-manifolds and links inside them.
• Just as an algebra is a category with one object, a monoidal category is a $2$-category with one object. Factorization homology shows that any n-algebra produces invariants of $n$-manifolds, and this can be generalized to $n$-algebras with "many objects", i.e., to $n$-categories.

Answer 3

Here is a fairly naive answer from a non-expert.

If you've studied algebraic topology then you can probably appreciate the philosophy that for the purposes of homotopy theory you should not consider just the category of topological spaces with continuous maps. The reason is that if two continuous functions are homotopic then they induce the same maps on homotopy groups and hence are "the same" for the purposes of homotopy theory. Something similar happens in algebraic geometry: two distinct morphisms might induce the same maps on cohomology.

One way to handle this is to pass from the category "Top" to the category "hTop" - the homotopy category. The objects in this category are topological spaces (actually, CW complexes) but the morphisms are homotopy equivalence classes of continuous functions. This is a great category for doing homotopy theory: for instance, if $X$ and $Y$ are connected and have the homotopy type of CW complexes and if $f, g \colon X \to Y$ are continuous functions which induce the same map on all homotopy groups then $f$ and $g$ are homotopic $f \colon X \to Y$ induces an isomorphism on all homotopy groups then $f$ is a homotopy equivalence (Whitehead's theorem).

But sometimes - particularly when it is useful to consider relationships between homotopy groups - the process of collapsing an entire equivalence class of continuous maps down to a single morphism is a bit too violent. Another way to proceed is, given homotopic maps $f$ and $g$, to make a choice $h_1$ of homotopy between them and remember it. But of course there are many possible choices of homotopies, some of which are homotopic (as homotopies from $f$ to $g$) to $h_1$ and some of which aren't. If you don't care about this distinction then you have a $2$-category: the objects are topological spaces, the morphisms are continuous functions, and the $2$-morphisms (morphisms between morphisms) are homotopy classes of homotopies between $f$ and $g$. If you do, then you can try to remember $f$ and $g$ as well as two homotopies $h_1$ and $h_1'$ from $f$ to $g$ and a homotopy $h_2$ from $h_1$ to $h_1'$. You can stop here and get a $3$-category, or you can keep going. If you never decide to stop, you get an $\infty$-category.

This kind of structure is particularly relevant in stable homotopy theory, where the main objects of studies are not spaces but spectra. A spectrum consists of an infinite family of topological spaces equipped with morphisms from the suspension of the $n$th space to the $n+1$st for each $n$; homotopy classes of maps from a space into and out of a spectrum correspond precisely (by the Brown representability theorem) to generalized cohomology and homology theories. Many spectra come equipped with a "smash product" which explains product and coproduct structures on cohomology and homology, and this product behaves a bit like the product in a ring. But in order to take seriously the algebraic structure of this product one must only insist that the ring axioms are satisfied up to homotopy, and indeed one must keep track of homotopies between homotopies and so on. This leads to the theory of $E_\infty$ ring spectra.

A big project in stable homotopy theory these days is to try to do "number theory" and "algebraic geometry" over $E_\infty$ ring spectra. This is leading to sophisticated new constructions in topology such as TMF (topological modular forms), though it is not clear that these constructions are yet proving theorems of interest to someone who works on, say, $3$-manifolds. This is probably what people who think the theory is "useless" really mean: there aren't a lot of applications outside the theory itself. In that sense it's a lot like many other subjects in mathematics: if you care about the examples then you should learn the theory, otherwise you should learn something else.

Answer 4

Re: David Corfield's comment, this answer will mostly address "higher as in $(\infty, 1)$-categories" rather than "higher as in $n$-categories."

I also do understand you need the notion of abelian categories for homological algebra and constructing sheaves like the grand project done by Grothendieck.

This homological story is all about sheaves of abelian groups, or maybe sheaves of chain complexes; these are "linear" phenomena. As explained in David Ben-Zvi's excellent answer to the closely related MO question linked to in the comments, you can think of higher categories / homotopical algebra as a "nonlinear" generalization of homological algebra, suited to answering questions like the following:

What is a sheaf of categories on a space? Or a sheaf of homotopy types? Or a sheaf of cohomology theories?

There are many motivating examples here in algebraic geometry and algebraic topology. Here is a basic one: let $X$ be a topological space, let $G$ be a discrete group, and consider the presheaf sending an open set $U$ to the set of isomorphism classes of principal $G$-bundles on $U$. This presheaf is not a sheaf: for example, if $X$ is a manifold, then sufficiently small open sets are contractible and contractible spaces admit a unique isomorphism class of principal $G$-bundle, but the classification of principal $G$-bundles on $X$ nevertheless detects the fundamental group of $X$.

The problem is that in order to construct principal $G$-bundles on a space from principal $G$-bundles on an open cover of that space, we need to keep track of isomorphisms themselves and not just the relation of being isomorphic. This is a basic, fundamental conceptual move in higher category theory. That is, rather than a presheaf of sets we should consider a presheaf of groupoids, or homotopy $1$-types, sending an open set $U$ to the groupoid of principal $G$-bundles on $U$.

This presheaf is a sheaf, in a suitable sense (it is a stack, but I think that term scares people so I'll stick to "sheaf" and not even say "$2$-sheaf" or anything). (I need some niceness condition on $X$ here; finite CW-complex should suffice.) Because groupoids live in a $2$-category rather than a category, the sheaf axiom needs to be modified, and in particular it is now necessary to talk about triple rather than just double intersections. The correct sheaf condition is now that if $U = \bigcup_i U_i$ is an open covering, then a suitable diagram of the form

$$F(U) \to \prod_i F(U_i) \Rightarrow \prod_{i, j} F(U_i \cap U_j) \Rrightarrow \prod_{i, j, k} F(U_i \cap U_j \cap U_k)$$

exhibits $F(U)$ as the 2-limit or homotopy limit of the rest of the diagram. Working out what this says about computing global sections from a nice open cover gets you precisely the Cech cocycle description of principal $G$-bundles.

(It's natural to want to allow $G$ to be a topological group here. Here we should really be keeping track of the entire space $[U, BG]$ of principal $G$-bundles on an open set $U$, and then the sheaf condition involves all finite intersections. But as a byproduct, by taking $BG$ to be an Eilenberg-MacLane space we recover the Cech cocycle description of higher cohomology.)

For an example closer to a modern direction of research, Lurie's approach to defining topological modular forms requires understanding what "the sheaf of elliptic cohomology theories over the moduli space of elliptic curves" means in a suitably derived sense, and then taking the global sections of this sheaf, again in a suitably derived sense.

For an example with relatively concrete geometric significance, one way to understand strong forms of the h-principle is that they answer questions of the following form: if $F, G$ are two presheaves of topological spaces on a manifold $X$ and $f : F \to G$ is a map of presheaves such that $F(U) \to G(U)$ is a (weak) homotopy equivalence for contractible open sets $U$, when can we conclude that $F(X) \to G(X)$ is a (weak) homotopy equivalence? (In particular, it does not suffice to assume that $F$ and $G$ are sheaves in the usual sense.) Motivating examples include the case that $F$ is the sheaf of immersions of $U$ into another manifold $Y$ and $G$ is the sheaf of "formal immersions": this leads to versions of the Hirsch-Smale theorem, as explained in these notes from a class by John Francis, that tell us some very concrete things about when manifolds can be immersed into, for example, $\mathbb{R}^n$.

(To be clear, I'm not claiming that you need to understand higher categories in order to understand the Hirsch-Smale theorem; certainly Hirsch and Smale did not need them. I just wanted to give an example where 1) people look at and care about presheaves of spaces and maps between them and 2) the usual sheaf condition doesn't answer the local-to-global question you want to answer.)

I have talked to several top mathematicians in my department, many of whom apperently think it is useless.

It's worth pointing out that mathematical culture differs greatly from department to department.

Answer 5

I refer you to a talk I gave in Paris last June on "The intuitions of cubical sets for nonabelian algebraic topology" which explains how from my 1967 groupoid version of the Seifert-van Kampen Theorem I was led to ask about the potential relevance of groupoids in higher homotopy theory. Eventually with Philip Higgins we obtained a new formulation of basic homology/homotopy theory using actual compositions of cubes, and avoiding Poincar'e's "formal sums", now expressed in terms of free abelian groups. The details are in the EMS Tract vol 15 (2011) advertised here, where there is a pdf available. This book also discusses the history and intuitions.

These methods give more understanding and also homotopical computations not available by other methods. As an indication, I mention the bibliography on the nonabelian tensor product which has 131 items, mainly in group theory. Relatively few algebraic topologists have worked in this area. Here is a short exposition related to the Blakers-Massey Theorem.

These papers use higher groupoids as concrete algebraic objects better able to express higher geometric ideas than purely $1$-dimensional constructions.

What is called "higher category theory" has not yet, it seems to me, accommodated the cubical approach. However for the work in which I have been involved the cubical methods have proved essential.

See also this mathoverflow answer on higher homotopy groupoids.