The n-Category Café

Given a space like $\mathbb{R}^n$ with coordinate functions $\{X^i\}$, we can form plenty of differential operators on the space of functions on this space by writing expressions in local coordinates like

or

The second one is “natural” (it comes from the Lie derivative), the first one isn’t.

What exactly does this mean?

Let $\mathrm{Man}_n$ be the category of $n$-dimensional smooth manifolds with morphisms being open embeddings. Let $\mathrm{Fib}_n$ be the category of fiber bundles over $\mathrm{Man}_n$.

We say a kind of bundle is natural if we can functorially associate it to manifolds:

Definition: A natural bundle is a functor $$\mathbf{F} : \mathrm{Man}_n \to \mathrm{Fib}_n$$ such that $F(M)$ is a bundle over $M$ and such that for any open submanifold $M' \subset M$ we have $F(M') = F(M)|_{M'}$.

In 1977 Palais and Terng proved a theorem which characterized natural bundles as precisely those fiber bundles that are associated by $\mathrm{GL}^k(n)$ to a $k$-frame bundle.

The bundle of $k$-frames over $M$

is defined to be the bundle of $k$-jets of local coordinate systems. This is like the ordinary bundle of frames plus higher derivatives of that. In particular, $F^1_n(M) = F_n(M)$ is the ordinary frame bundle of $M$.

Similarly, $\mathrm{GL}^k(n)$ is the group of $k$-jets of local diffeomorphisms of $M$.

So:

Theorem: For each natural bundle $\mathbf{F}$ there is a natural number $k \geq 1$ and a $\mathrm{GL}^k(n)$-space $F$ such that

For instance, the tangent bundle is natural and we have

Now,

Definition:A natural differential operator is any morphism $$D : \mathbf{F}^{(l)} \to \mathbf{G} \,,$$ where $\mathbf{F}$ and $\mathbf{G}$ are natural bundles and $\mathbf{F}^{(l)}$ is the natural bundle obatined by taking $l$-jets of $\mathbf{F}$.

For instance, the Lie derivative takes two copies of the tangent bundle to the tangent bundle, depending on the first derivative of the vector fields involved. Hence it is a natural operator of the form

The nice thing is that such natural differential operators can be understood in terms of equivariant maps:

Theorem: Let $\mathbf{F}$ and $\mathbf{G}$ be natural bundles with typical fibers $F$ and $G$, respectively, according to the above theorem. Then we have a bijection between natural differential operators $$D : \mathbf{F}^{(l)} \to \mathbf{G}$$ and $\mathrm{GL}^{k+l}(n)$-equivariant maps $$U : F^{(l)} \to G \,.$$

$U$ is the local formula for the differential operator. For a reason unknown to the speaker and his audience, this theorem is known as IT-reduction.

Next, we can decompose $F^{(l)}$ as well as $G$ into reps of the ordinary general linear group $\mathrm{GL}(n)$. The basic invariant theorem for such reps then tells us that all natural differential operators must be obtainable by doing the familiar index contraction on linear maps, roughly.

But we don’t want to think in terms of index contraction, but instead in terms of plumbing. A linear map is represented by a graph with a single vertex, with a couple of incoming and a couple of outgoing edges. Index contraction is conneting outgoing with incoming edges between maps.

The main message is that one can define a differential $\delta$ on graphs, which acts locally - in the sense that its action is completely specified by the action on graphs containing a single vertex - such that the natural differential operators come from precisely those linear maps $U$ such that

Markl indicated that there is something much more profound going on in the background, involving operads. But that’s where the talk ended.