# nLab path n-groupoid

Contents

### Context

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Idea

A path $n$-groupoid $P_n(X)$ of a smooth space (or generalized smooth space) $X$ is a diffeological n-groupoid which is

• a generalization of the path groupoid to higher categorical dimension

• a truncation of sorts of an path ∞-groupoid of $X$.

Its j-morphisms are given by (possibly equivalence classes of) $j$-dimensional smooth paths in $X$, i.e. usually smooth maps $\gamma : D^j \to X$. Composition is by gluing of such maps.

## Path 1-groupoid

See path groupoid.

## Path 2-groupoid

Definitions of path 2-groupoids as strict 2-groupoids internal to diffeological spaces appear (at least) in

For the underlying notion of fundamental 2-groupoid see there.

## Path 3-groupoid

A realization of the path 3-groupoid as a Gray-groupoid internal to diffeological spaces appears in

Last revised on January 19, 2023 at 11:42:35. See the history of this page for a list of all contributions to it.