# nLab Lie algebroid-groupoid

Contents

### Context

#### Higher category theory

higher category theory

## Operadic case

#### 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

tangent cohesion

differential cohesion

graded differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## Idea

An internal groupoid in the category of Lie algebroids.

## Examples

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

$\array{ T \mathcal{G}_1 &\to& \mathcal{G}_1 \\ {}^{\mathllap{d s}}\downarrow \downarrow^{\mathrlap{d t}} && {}^{\mathllap{s}}\downarrow \downarrow^{\mathrlap{t}} \\ T \mathcal{G}_0 &\to& \mathcal{G}_0 } \,.$

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