# nLab foliation of a Lie groupoid

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

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 $X$ is a wide sub-Lie algebroid of its tangent Lie algebroid, hence a Lie algebroid $\mathcal{P}$ over $X$ with injective anchor map

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

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 $\mathcal{G}_\bullet$ is a sub-Lie algebroid-groupoid of the tangent Lie algebroid-groupoid which is wide

$\array{ \mathcal{P} &\hookrightarrow& T \mathcal{G}_1 \\ \downarrow \downarrow && \downarrow \downarrow \\ \mathcal{P}_0 &\hookrightarrow& T \mathcal{G}_0 } \,.$

## Examples

(…)

Maybe the first discussion of foliations of Lie groupoids appears in

Related discussion is in