Natural operators in differential geometry - why are they natural?

Question

I'm reading bits and pieces of Kolar, Michor, & Slovak's Natural Operations in differential Geometry, and I'm having "doubt" about some of the definitions. All I'm trying to do is sheafify some of the concepts and see how natural they seem to me. I don't know any differential geometry so please don't kill me.

Let $l\mathsf{Diff}_m$ denote the category of smooth manifolds and local diffeomorphisms. A fibered manifold is a surjective submersion.

Definition 14.1 A natural bundle is a functor $l\mathsf{Diff}_m\rightarrow \mathsf{FM}$ satisfying

• Prolongation $BF=1$ where $B$ is the base functor sending a fibered manifold into its base space.
• Locality - If $i:U\hookrightarrow M$ is an inclusion of an open submanifold, then $FU=p^{-1}_M (U)$ and $Fi:FU\hookrightarrow FM$ is the inclusion of $FU$ into $FM$.

Denote by $\Gamma(Y)$ the (global) sections of a fibered manifold $p:Y\rightarrow M$.

Definition 14.13 Let $p:Y\rightarrow M,\bar p:\bar Y\rightarrow M$ be fibered manifolds. A local operator $A:\Gamma(Y)\rightarrow \Gamma(\bar Y)$ is a map such that for every global section $s$ and every point $x\in M$, the value of $As(x)$ depends only on the germ of $s$ at $x$. If moreover, for some $k\in \mathbb N$ we have $j_x^ks=j^k_sq\implies As(x)=Aq(x)$, $A$ is said to be of order $k$. A regular operator is a local operator which sends smoothly parametrized section into smoothly parametrized sections into smoothly parametrized sections.

First batch: So it seems a local operator is just a map $A:\Gamma(Y)\rightarrow \Gamma(\bar Y)$ which lifts to stalks $A_x:\Gamma(Y)_x\rightarrow \Gamma(\bar Y)$. Do we get a lift $\Gamma(Y)_x\rightarrow \Gamma(\bar Y)_x$?

For the following definition, it seems the value of a natural bundle $F$ at $N$ is a bundle $FN\rightarrow N$ and that we no longer fix a base space $M$.

Definition 14.15 A natural operator $A:F\rightsquigarrow G$ between two natural bundles $F$ and $G$ is a system of regular operators $A_M:\Gamma(FM)\rightarrow \Gamma(GM),\;M\in l\mathsf{Diff}_m$ satisfying

• For each global section and each diffeomorphism we have $A_N(Ff\circ s\circ f^{-1})=Gf\circ A_Ms\circ f^{-1}$
• $A_U(s|_U)=(A_Ms)|_U$ for each global section $s$ and every open submanifold $U\subset M$

Second batch: First of all, why is $f$ taken to be a diffeo? Shouldn't we ask for commutation for all local diffeos? They're the arrows in our category after all... If so, then it seems a natural operator is natural in two different ways: Given a natural bundle $F$, define the functor $\Gamma(-,F-):l\mathsf{Diff}_m\longrightarrow \mathsf{Set}$ on objects by global sections, and on arrows by $\Gamma(f,Ff):s\mapsto Ff\circ s\circ f^{-1}$. In fact, each $\Gamma(M,FM)$ is really the sheaf of sections of the bundle $FM\rightarrow M$, so $\Gamma(-,F-)$ seems to yield a functor taking values in sheaves (I don't know into what category one should stick all sheaves). Now it seems a natural operator is a natural transformation $A:\Gamma(-,F-)\Rightarrow \Gamma(-,G-)$ which also respect the sheaf structure in that the components $(A_U)$ for open submanifolds $U$ of a fixed manifold $M$ also give a sheaf morphism $\Gamma(M,FM)\Rightarrow \Gamma(M,GM)$. Assuming I am not too far off, and that this really is equivalent data to a natural operator, I have to admit that this notion doesn't seem all that natural to me at all! Could someone geometrically motivate this notion? Why shouldn't we be satisfied with mere natural transformations $F\Rightarrow G$?

Answer 1

This is a long comment on some of the questions from the second batch. Before I start, here is a typical example of a natural operator. Take $F=\Lambda^kT^*$, $G=\Lambda^{k+1}T^*$ to be the natural bundles of differential forms of degrees $k$ and $k+1$. They are clearly local, and they behave naturally under pullback (which is not asked for here). But local diffeos have local inverses, by which we pull back, so we can regard them as covariant functors as in Definition 14.1 (this part looks unnatural to me in a nontechnical sense, but the OP did not complain about this point).

The exterior differential $d$ is a natural operator sending $\alpha\in\Gamma(\Lambda^kT^*N)$ to $d\alpha\in\Gamma(\Lambda^{k+1}T^*N)$. Again, it is local, and compatible with pullback by smooth maps. In particular, it is compatible with pullback by inverses of global diffeomorphisms, which is the first point in Definition 14.15.

If you are used to think in terms of sheafs, you want to rewrite 14.15 in terms of sheafs, and you will get an equivalent notion. However, some differential geometers don't linke to think in sheafs, but rather in terms like "local" and "natural under diffeomorphisms". So up to the very last few sentences, I think you understood everything perfectly.

Now to the last remark. In the example above, you are not allowed to write $d\colon\Lambda^kT^*-\Rightarrow\Lambda^{k+1}T^*-$ because elements of $\Lambda^kT^*M$ are differential $k$-forms at some single point $x\in M$ (not germs), so you cannot differentiate. Indeed, a natural operator in the sense above can be written as a natural transformation $F-\Rightarrow G-$ if and only if it is of degree $0$.

Answer 2

2nd batch: What is $f^{-1}$ for a local diffeo. Locality is encoded in the second condition. Of course you can translate into the language of sheaves - but is not necessary.

1st batch: A local operator lifts to stalks.

Note: a linear local operator (between spaces of sections of vector bundles) is a finite order differential operator (thm of Peetre) A (nonlinear) local operator is a differential operator (which might be of infinite order at discrete points (thm of J. Slovak, called the nonlinear Peetre theorem)

A natural operator is always of finite order. Looking at its jet prolongations allows for classification.