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 14.1Anatural bundleis 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 14.13Let $p:Y\rightarrow M,\bar p:\bar Y\rightarrow M$ be fibered manifolds. Alocal 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 oforder$k$. Aregular operatoris 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 14.15Anatural 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 $F\Rightarrow G$?