vector bundle, 2-vector bundle, (∞,1)-vector bundle
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(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A type of fiber bundle (usually: vector bundle) over (smooth) manifolds is called natural if suitable homomorphisms of manifolds naturally lift to morphisms of the corresponding bundles covering them.
The archetypical example is the tangent bundle $T X$ of a manifold $X$. For $f \colon X \to Y$ any smooth function then push-forward of vector fields yields a fiberwise linear morphism $d f \colon T X \to T Y$ of tangent bundles which covers $f$, in that the diagram
commutes. (This functoriality of the tangent bundle construction is incidentally also the incarnation of the chain rule (see there) of differentiation).
Formally, there is a category of fiber bundles over manifolds with morphisms the morphisms of bundles covering morphisms of the base spaces, and there is a projection functor from fiber bundles to their base manifolds. A natural bundle is a section of this projection functor.
Also “contravariantly natural” bundles such as bundle of covectors are sometimes referred to as natural bundles (e.g. Kolář-Michor-Slovak). But these are covariantly natural not with respect to all smooth functions between manifolds, but only for local diffeomorphisms.
Originally introduced in
Albert Nijenhuis, Geometric aspects of formal differential operations on tensor fields, Proceedings of the International Congress of Mathematicians 1958, 463–469. PDF.
Albert Nijenhuis, Natural bundles and their general properties, in: Differential Geometry (in honor of Kentaro Yano), Kinokuniya, Tokyo, 1972, pp. 317–334. PDF.
A comprehensive reference is available in
The example of the Schouten-Nijenhuis bracket:
Last revised on May 4, 2023 at 16:47:57. See the history of this page for a list of all contributions to it.