nLab
jet groupoid

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

Given a smooth manifold XX, and given k{}k \in \mathbb{N}\cup \{\infty\}, there is a Lie groupoid whose objects are the points of XX and whose morphisms between two such points xyx\to y are order-kk jets of local diffeomorphisms taking xx to yy.

The automorphism groups of objects in these groupoids are jet groups.

Hence these Lie groupoids are often called jet groupoids, e.g. (Lorenz 09). If one passes from jets to germs of local diffeomorphisms then one arrives essentially at the Haefliger groupoid of XX (except that this has a more discrete smooth structure on its set of morphisms).

In a context of synthetic differential geometry or differential cohesion there is the bundle T inf kXXT_{inf}^k X\to X of order-kk infinitesimal neighbourhoods in XX. In terms of this the jet groupoid is the Atiyah groupoid of T inf kXT_{inf}^k X, the groupoid whose morphisms between objects xx and yy are isomorphism of the fibers of this bundle over these points.

References

  • Arne Lorenz, Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method, Thesis (pdf) 2009

Last revised on January 4, 2015 at 23:47:15. See the history of this page for a list of all contributions to it.