Let $\mathrm{Bundle}$ be the category whose objects are [smooth vector fiber bundles](https://en.wikipedia.org/wiki/Fiber_bundle) over $\mathbb{R}$, and morphisms are fiberwise smooth linear map (that is, the base is not assumed to be fixed).


Let $Base \colon \mathrm{Bundle} \to \mathrm{Diff}$ be the functor returning the base of a given bundle (and the morphism between bases of a given bundle morphism, respectively). Let's define $ \mathrm{FunctorialBundle}$ as the full subcategory of $ \mathrm{Func}(\mathrm{Diff}, \mathrm {Bundle})$ on those functors that are section to $Base$ (that is, $ F \in \mathrm{Ob}~\mathrm{FunctorialBundle} \ \Leftarrow:\Rightarrow\ F \circ Base = id $).


**Question 1**: Classify objects of $ \mathrm{FunctorialBundle} $ up to isomorphism.


Of course, my question is rather "is such a classification possible?". Or is the class of such functors immense and the classification problem does not make sense (just as the problem of classifying all finite magmas, semigroups, groups does not make sense)? My next question is an example of an answer:


Starting with tangent bundle $TM$ and one-dimensional trivial bundle $\mathbb{R} \times M$ and applying operations direct sum, tensor product, dual we obtain all tensor bundle functors (whose sections are tensor fields). If there are some standard operations that extend the functor class, then add them to this list.


**Question 2**: are there functorial vector bundles (that is, objects of $ \mathrm{FunctorialBundle} $ ) that are not in this class (up to isomorphism, of course)?



Classification of functorial smooth vector fiber bundles
This is called [natural bundle](https://ncatlab.org/nlab/show/natural+bundle). Apparently, all known information is in [Kolár, Slovák, Michor: Natural operations in differential geometry](https://www.mat.univie.ac.at/%7Emichor/kmsbookh.pdf) (recommended by Stefan Waldmann).


From the description, it is also the best categorical textbook on differential geometry and topology.



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> *Third* in the beginning of this book we try to give an introduction to the fundamentals of differential geometry (manifolds, flows, Lie groups, differential forms, bundles and connections) which stresses naturality and functoriality from the beginning and is as coordinate free as possible. Here we present the Frölicher–Nijenhuis bracket (a natural extension of the Lie bracket from vector fields to vector valued differential forms) as one of the basic structures of differential geometry, and we base nearly all treatment of curvature and Bianchi identities on it. This allows us to present the concept of a connection first on general fiber bundles (without structure group), with curvature, parallel transport and Bianchi identity, and only then add G-equivariance as a further property for principal fiber bundles. We think, that in this way the underlying geometric ideas are more easily understood by the novice than in the traditional approach, where too much structure at the same time is rather confusing.
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