nLab
foliation of a Lie 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)

Higher Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

A generalization of the notion of foliation of a smooth manifold from manifolds to Lie groupoids.

Definition

One of several equivalent definitions of a (regular) foliation of a smooth manifold is

Definition

A regular foliation of a smooth manifold XX is a wide sub-Lie algebroid of its tangent Lie algebroid, hence a Lie algebroid 𝒫\mathcal{P} over XX with injective anchor map

𝒫 TX X = X. \array{ \mathcal{P} &\hookrightarrow & T X \\ \downarrow && \downarrow \\ X &=& X } \,.

In this spirit there is an evident generalization of the notion to a notion of foliations of Lie algebroids.

Definition

A (regular) foliation of a Lie groupoid 𝒢 \mathcal{G}_\bullet is a sub-Lie algebroid-groupoid of the tangent Lie algebroid-groupoid which is wide

𝒫 T𝒢 1 𝒫 0 T𝒢 0. \array{ \mathcal{P} &\hookrightarrow& T \mathcal{G}_1 \\ \downarrow \downarrow && \downarrow \downarrow \\ \mathcal{P}_0 &\hookrightarrow& T \mathcal{G}_0 } \,.

Examples

(…)

Referemces

Maybe the first discussion of foliations of Lie groupoids appears in

Related discussion is in

Created on March 25, 2013 at 22:24:04. See the history of this page for a list of all contributions to it.