The n-Category Café

The splitting principle

The splitting principle is a fundamental trick in algebraic topology, representation theory, and algebraic geometry. It lets us study a complicated object by finding a larger category in which this object splits as a direct sum of simpler ones. For example, suppose you have a vector bundle $E$ over a connected compact manifold $X$. It’s usually not a direct sum of line bundles… but using the splitting principle, you can sometimes pretend it is!

The reason is that there’s some other compact manifold $Y$ and a map $\phi \colon Y \to X$ such that the pullback $\phi^\ast E$, a vector bundle over $Y$, is a direct sum of line bundles. So far this is obvious: for example, we can take $Y$ to be the empty set! But in fact, we can choose $Y$ so that the pullback map $\phi^\ast$ is ‘one-to-one’ in a certain sense. This lets us reason about vector bundles as if they were direct sums of line bundles, and get valid conclusions.

Let me spell this out a bit, and generalize the idea using some category theory.

2-Rigs

For any compact manifold $X$ there’s a category of vector bundles $\mathsf{Vect}(X)$ over $X$. But this much better than a mere category! We can add vector bundles by taking their direct sum, $\oplus$, and multiply them by taking their tensor product $\otimes$. Tensor product distributes over direct sum, so $\mathsf{Vect}(X)$ is like a categorified ring. But it doesn’t have negatives, so it’s really a categorified rig.

We can make this precise using a specific notion of categorified rig. For the purposes of our paper, we define a 2-rig to be a Cauchy complete symmetric monoidal $k$-linear category where $k$ is a field of characteristic zero. Here a $k$-linear category is Cauchy complete if:

• It has biproducts. These are products and coproducts in a compatible way. In a category of vector bundles, these are just what we call direct sums of vector bundles.

• We can split idempotents. This means that given any endomorphism $p: c \to c$ with $p^2 = p$, we can write $c = c_1 \oplus c_2$ in such a way that $p$ is the projection $c_1 \oplus c_2 \to c_1$ followed by the inclusion $c_1 \to c_1 \oplus c_2$. We can always do this in a category of vector bundles: any idempotent map from a vector bundle to itself is the projection onto some sub-bundle that is a summand of our original bundle.

There are many examples of 2-rigs: not only categories of vector bundles, but also categories of group representations, or sheaves of vector spaces, or coherent sheaves of vector spaces. Our paper develops many more!

Grothendieck rings

There’s a way to go from the world of 2-rigs to the more familiar world of rings. Any 2-rig $\mathsf{R}$ gives a commutative ring $K(\mathsf{R})$ called its Grothendieck ring. To construct this, we first take the set of isomorphism classes of objects of $\mathsf{R}$. This becomes a rig if we define

$$[r] + [s] = [r \oplus s]$$

$$[r] \, [s] = [r \otimes s]$$

We then throw in formal additive inverses, turning this rig into the ring we call $K(\mathsf{R})$.

But $K(\mathsf{R})$ is more than a mere commutative ring! We can define exterior powers $\Lambda^d r$ of any object $r$ in a 2-rig. We can do this by splitting idempotents. This makes $K(\mathsf{R})$ into a ‘$\lambda$-ring’, meaning that it has operations

$$\lambda^d \colon [r] \mapsto [\Lambda^d r]$$

one for each $d = 0, 1, 2, \dots$, obeying a bunch of standard identities. Indeed, my impression is that lambda-rings were developed by Grothendieck around the same time he was developing the splitting principle in algebraic geometry.

The splitting principle, revisited

Any map $\phi \colon Y \to X$ between compact manifolds lets us pull back vector bundles, giving a functor

$$\phi^\ast \colon \mathsf{Vect}(X) \to \mathsf{Vect}(Y)$$

This is a map of 2-rigs: a symmetric monoidal $k$-linear functor. A nice fact is that any map of 2-rigs

$$f \colon \mathsf{R} \to \mathsf{R'}$$

automatically preserves biproducts and splittings of idempotents. Furthermore, any map of 2-rigs induces a $\lambda$-ring homomorphism between their Grothendieck rings, which we call

$$K(f) \colon K(\mathsf{R}) \to K(\mathsf{R'})$$

Now I can state a more precise version of the splitting principle (there are many):

A Splitting Principle for Vector Bundles. If $X$ is a connected compact manifold, for any vector bundle $E \in \mathsf{Vect}(X)$ we can find a compact manifold $Y$ and a map $\phi \colon Y \to X$ such that

1. $\phi^\ast(E)$ is a direct sum of finitely many line bundles.

2. $K(\phi^\ast) \colon K(\mathsf{Vect}(X)) \to K(\mathsf{Vect}(Y))$ is injective.

3. $\phi^\ast \colon \mathsf{Vect}(X) \to \mathsf{Vect}(Y)$ is a 2-rig extension, meaning it is faithful, conservative (i.e. it reflects isomorphisms), and essentially injective.

Condition 1 is the main idea: we can carry our bundle $E$ over to a new world where it’s a sum of line bundles.

But condition 2 is also important. It eliminates silly choices like $Y = \emptyset$. More importantly, it means that if we have a vector bundle $E$ over $X$, and we want to prove some equation involving its class in $K(\mathsf{Vect}(X))$, we can often pretend $E$ splits as a sum of line bundles. To do this, we switch to looking at $\phi^\ast(E)$, which does split as a direct sum of line bundles. We show our equation holds for $\phi^\ast(E)$, over in $K(\mathsf{Vect}(Y))$. Then, since $K(\phi^\ast)$ is injective, we can often get the equation we originally wanted, back in $K(\mathsf{Vect}(X))$.

Condition 3 is a list of three other ways in which $\phi^\ast$ is injective. These are not part of the traditional splitting principle, but they are crucial if we want to categorify the splitting principle.

Why did I assume $X$ is connected above? Because we need to! Here’s a counterexample if we drop the connectness assumption. Suppose $X$ has two components and the vector bundle $E$ over $X$ has 1-dimensional fibers over one component and 0-dimensional fibers over the other. This will cause trouble.

You may want to rule out weird vector bundles like this. But then you’ll suffer in other ways. It’s better to introduce the concept of a subline bundle: a sub-vector-bundle of a line bundle. The vector bundle I just mentioned is not a line bundle, but it’s a subline bundle.

Using this concept we get a more general splitting principle:

Improved Splitting Principle for Vector Bundles. If $X$ is a compact manifold, for any vector bundle $E \in \mathsf{Vect}(X)$ we can find a compact manifold $Y$ and a map $\phi \colon Y \to X$ such that

1. $\phi^\ast(E)$ is a direct sum of finitely many subline bundles.

2. $K(\phi^\ast)$ is injective.

3. $\phi^\ast$ is a 2-rig extension.

Generalizing the splitting principle to 2-rigs

With these lessons in hand, let’s try to generalize the splitting principle to other 2-rigs.

First, let’s say an object $r$ in a 2-rig has subdimension $d$ if $\Lambda^{d+1} r \cong 0$. For example, a vector space has subdimension $d$ iff its dimension is $\le d$.

Second, let’s call an object of subdimension 1 a subline object. For example, a vector bundle is a subline object in $\mathsf{Vect}(X)$ if and only if it’s subline bundle.

Now we’re ready to make a conjecture:

Conjecture: The Splitting Principle for 2-Rigs. Let $\mathsf{R}$ be a 2-rig and $r \in \mathsf{R}$ an object of finite subdimension. Then there exists a 2-rig $\mathsf{R}'$ and a map of 2-rigs $f \colon \mathsf{R} \to \mathsf{R}'$ such that:

1. $f(r)$ splits as a direct sum of finitely many subline objects.

2. $K(f) \colon K(\mathsf{R}) \to K(\mathsf{R}')$ is injective.

3. $f \colon \mathsf{R} \to \mathsf{R}'$ is a 2-rig extension.

The free 2-rig on one object

We prove a version of the above conjecture for the universal example, namely the free 2-rig on one object. This beautiful entity is often called the category of Schur functors, but we denote it by the scary name $\overline{k \mathsf{S}}$ since it can be obtained by the following three-step process:

• First form the free symmetric monoidal category on one generating object, say $x$. This is equivalent to groupoid of finite sets and bijections, which we call $\mathsf{S}$, with disjoint union providing the symmetric monoidal structure. Up to equivalence, $\mathsf{S}$ has one object for each natural number $n$, and the endomorphisms of this object form the symmetric group $S_n$. So you could call it the symmetric groupoid: it’s a way of studying all the symmetric groups together at once.

• Then form the free $k$-linear symmetric monoidal category on $\mathsf{S}$ by freely forming $k$-linear combinations of morphisms. This is called $k\mathsf{S}$. Up to equivalence, it has one object for each natural number $n$, and the endomorphisms of this object form an algebra, the group algebra of $S_n$.

• Then Cauchy complete $k\mathsf{S}$: that is, take direct sums of objects and split idempotents. The result is a 2-rig called $\overline{k \mathsf{S}}$. As a $k$-linear category this is the coproduct of the categories of finite-dimensional representations of all the symmetric groups $S_n$. Thus, it’s a way of studying the representation theory of all the symmetric groups together at once.

Now, the generating object $x \in \overline{k \mathsf{S}}$ does not have finite subdimension! Since this 2-rig is free, none of the exterior powers $\Lambda^d x$ vanish. So, for our splitting principle, we should extend $\overline{k \mathsf{S}}$ to a larger 2-rig where $x$ splits as an infinite direct sum of subline objects.

To do this, we define a 2-rig map

$$f \colon \overline{k \mathsf{S}} \to \mathsf{A}^{\boxtimes \infty}$$

from $\overline{k \mathsf{S}}$ to the limit

$$\mathsf{A}^{\boxtimes \infty} := \lim_{\longleftarrow} \; \mathsf{A}^{\boxtimes N}$$

where $\mathsf{A}^{\boxtimes N}$ is our name for the free 2-rig on $N$ subline objects, say $s_1, \dots, s_N$. The 2-rig $\mathsf{A}^{\boxtimes \infty}$ contains infinitely many subline objects $s_1, s_2, s_3, \dots$. The 2-rig map $f$ is characterized by the fact that it sends the generating object $x \in \overline{k \mathsf{S}}$ to the infinite coproduct $s_1 \oplus s_2 \oplus \cdots$.

Here is our new splitting principle:

Theorem. The 2-rig map $f \colon \overline{k \mathsf{S}} \to \mathsf{A}^{\boxtimes \infty}$ has these properties:

1. $f(r)$ splits as a countably infinite coproduct of subline objects.

2. $K(f)$ is injective.

3. $f$ is a 2-rig extension.

This categorifies some classical results. In our previous paper we showed that because $\overline{k \mathsf{S}}$ is the free 2-rig on one generator, $K(\overline{k \mathsf{S}})$ is the free $\lambda$-ring on one generator. Here we show $K(A^{\boxtimes \infty})$ is the subring of $\mathbb{Z}[[x_1, x_2, \dots]]$ consisting of formal power series that are of bounded degree: each subline object $s_i$ gives a variable $x_i$. Since

$$K(f) \colon K(\overline{k \mathsf{S}}) \to K(\mathsf{A}^{\boxtimes \infty})$$

is an injective map of $\lambda$-rings, its image must be the free $\lambda$-ring on one generator. But its image consists precisely of symmetric functions: formal power series in $x_1, x_2, \dots,$ that are of bounded degree and invariant under all permutations of the variables.

So, the free $\lambda$-ring on one generator is isomorphic to the ring of symmetric functions!

Another taste of topology

It’s worth looking at how these results are connected to the splitting principle for vector bundles. The classifying space $\mathrm{BU}$ of the infinite-dimensional unitary group

$$\mathrm{U} = \lim_{\longrightarrow} \; \mathrm{U}(n)$$

has the property that $K(\mathrm{BU})$ is the free $\lambda$-ring on one generator. But if one defines a subgroup $\mathrm{T} \subset \mathsf{U}$ by

$$\displaystyle{ \mathrm{T} = \lim_{\longrightarrow} \; \mathrm{T}^n }$$

where the torus $\mathrm{T}^n$ consists of the diagonal matrices in $\mathrm{U}(n)$, then one can show $K(\mathrm{BT})$ is the subring of $\mathbb{Z}[[x_1, x_2, \dots ]]$ consisting of power series of bounded degree. Furthermore, the inclusion of $\mathrm{T}$ in $\mathrm{U}$ gives a map $\phi \colon \mathrm{BT} \to \mathrm{BU}$ for which the map $K(\phi) \colon K(\mathrm{BU}) \to K(\mathrm{BT})$ is injective. Its image is the ring of symmetric functions!

In this situation we can also take the universal $n$-dimensional vector bundle over $\mathrm{BU}$, pull it back along $\phi$, and split the resulting bundle into a sum of line bundles whose classes in $K(\mathrm{BU})$ are $x_1, \dots, x_n$.

Further results and conjectures

Hazewinkel has written of $\Lambda$ that

It seems unlikely that there is any object in mathematics richer and/or more beautiful than this one.

But the $\lambda$-ring structure on symmetric functions is often described using rather complicated and unintuitive formulas. As I’ve indicated, our theorem above lets us understand this $\lambda$-ring structure more conceptually, and see why symmetric functions form the free $\lambda$-ring on one generator.

However, this was not our main goal. More important was to categorify this result, and arguably still more important is to set the categorified result into a broader theory of 2-rig extensions. Indeed, to understand the splitting principle, we need to prove results about various other interesting topics:

• 2-rigs of representations of affine categories: that is, categories enriched in affine schemes.
• The 2-rig of representations of the affine monoid $\mathrm{M}(n,k)$ consisting of $n \times n$ matrices with entries in $k$. We conjecture this is the free 2-rig on one object of subdimension $n$.
• The 2-rig of representations of the affine group $\mathrm{GL}(n,k)$ consisting of $n \times n$ matrices with entries in $k$. We conjecture this is the free 2-rig on one object of dimension $n$.
• Graded 2-rigs. I said that $\mathsf{A}^{\boxtimes \infty}$ is the limit, as $N \to \infty$, of the free 2-rig on $N$ subline objects. But this is only true in the category of graded 2-rigs. If we took the limit in the category of 2-rigs, we’d get something much larger, and $K$ of this would probably be $\mathbb{Z}[[x_1, x_2, \dots]]$ instead of what we want: formal power series of bounded degree.

In short, we develop a lot of categorified algebra as a way to rethink the splitting principle and its connections to representation theory.