nLab Lie algebroid-groupoid

Contents

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Internal categories

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

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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}{}& \esh &\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

An internal groupoid in the category of Lie algebroids.

Examples

For 𝒢 \mathcal{G}_\bullet a Lie groupoid, forming degreewise the tangent Lie algebroid yields the tangent Lie algebroid groupoid

T𝒢 1 𝒢 1 ds dt s t T𝒢 0 𝒢 0. \array{ T \mathcal{G}_1 &\to& \mathcal{G}_1 \\ {}^{\mathllap{d s}}\downarrow \downarrow^{\mathrlap{d t}} && {}^{\mathllap{s}}\downarrow \downarrow^{\mathrlap{t}} \\ T \mathcal{G}_0 &\to& \mathcal{G}_0 } \,.

References

The notion is considered (under the name “ℒ𝒜\mathcal{L A}-groupoids”) in

Last revised on February 20, 2016 at 12:04:14. See the history of this page for a list of all contributions to it.