nLab
path n-groupoid

Context

\infty-Lie theory

∞-Lie theory

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 path nn-groupoid P n(X)P_n(X) of a smooth space (or generalized smooth space) XX is a diffeological n-groupoid which is

Its j-morphisms are given by (possibly equivalence classes of) jj-dimensional smooth paths in XX, i.e. usually smooth maps γ:D kX\gamma : D^k \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

Path 3-groupoid

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

Revised on September 4, 2010 14:38:33 by Urs Schreiber (134.100.32.213)