path infinity-groupoid

**∞-Lie theory** (higher geometry)
## Background
### Smooth structure
* generalized smooth space
* smooth manifold
* diffeological space
* Frölicher space
* smooth topos
* Cahiers topos
* smooth ∞-groupoid, concrete smooth ∞-groupoid
* synthetic differential ∞-groupoid
### Higher groupoids
* ∞-groupoid
* groupoid
* 2-groupoid
* strict ∞-groupoid
* crossed complex
* ∞-group
* simplicial group
### Lie theory
* Lie theory
* Lie integration, Lie differentiation
* Lie's three theorems
* Lie theory for stacky Lie groupoids
## ∞-Lie groupoids
* ∞-Lie groupoid
* strict ∞-Lie groupoid
* Lie groupoid
* differentiable stack
* orbifold
* ∞-Lie group
* Lie group
* simple Lie group, semisimple Lie group
* Lie 2-group
## ∞-Lie algebroids
* ∞-Lie algebroid
* Lie algebroid
* Lie ∞-algebroid representation
* L-∞-algebra
* model structure for L-∞ algebras: on dg-Lie algebras, on dg-coalgebras, on simplicial Lie algebras
* Lie algebra
* semisimple Lie algebra, compact Lie algebra
* Lie 2-algebra
* strict Lie 2-algebra
* differential crossed module
* Lie 3-algebra
* differential 2-crossed module
* dg-Lie algebra, simplicial Lie algebra
* super L-∞ algebra
* super Lie algebra
## Formal Lie groupoids
* formal group, formal groupoid
## Cohomology
* Lie algebra cohomology
* Chevalley-Eilenberg algebra
* Weil algebra
* invariant polynomial
* Killing form
* nonabelian Lie algebra cohomology
## Homotopy
* homotopy groups of a Lie groupoid
## Related topics
* ∞-Chern-Weil theory
## Examples
### $\infty$-Lie groupoids
* Atiyah Lie groupoid
* fundamental ∞-groupoid
* path groupoid
* path n-groupoid
* smooth principal ∞-bundle
### $\infty$-Lie groups
* orthogonal group
* special orthogonal group
* spin group
* string 2-group
* fivebrane 6-group
* unitary group
* special unitary group
* circle Lie n-group
* circle group
### $\infty$-Lie algebroids
* tangent Lie algebroid
* action Lie algebroid
* Atiyah Lie algebroid
* symplectic Lie n-algebroid
* symplectic manifold
* Poisson Lie algebroid
* Courant Lie algebroid
* generalized complex geometry
### $\infty$-Lie algebras
* general linear Lie algebra
* orthogonal Lie algebra, special orthogonal Lie algebra
* endomorphism L-∞ algebra
* automorphism ∞-Lie algebra
* string Lie 2-algebra
* fivebrane Lie 6-algebra
* supergravity Lie 3-algebra
* supergravity Lie 6-algebra
* line Lie n-algebra

The path ∞-groupoid $\Pi(X)$ of a generalized smooth space $X$ is a smooth version of the fundamental ∞-groupoid of $X$. Its truncations to lower categorical degree yield

One way to define a path ∞-groupoid in terms of Kan complexes is to let

$\Delta_{SmoothSp} : \Delta \to SmoothSp$

be the canonical cosimplicial object in smooth spaces that sends the abstract $n$-simplex $\Delta[n]$ to the standard smooth $n$-simplex $\Delta^n \subset \mathbb{R}^n$.

As every cosimplicial object with values in a category with colimits this induces a notion of nerve and realization. The smooth nerve operation

$N : SmoothSp \to SmoothSp^{\Delta^{op}}$

with values in smooth ∞-stacks given by

$N(X) : U \mapsto SmoothSp(U \times \Delta^\bullet_{SmoothSp}, X)
\,,$

where on the right we have a simplicial object in the category of smooth spaces regarded as a model for a smooth ∞-stack.

Notice that the Kan complex valued sheaf presented by this is given for instance by the simplicial sheaf

$N(X) : U \mapsto Ex^\infty SmoothSp(U \times \Delta^\bullet_{SmoothSp}, X)
\,,$

which can be thought of as having in degree $k$ *piecewise smooth* $k$-dimensional paths.

Functors out of the path groupoid and path n-groupoid represent connections and higher connectios. Discussion of this for the path $\infty$-groupoid is here.

A more detailed account is at

Revised on December 21, 2009 14:00:01
by Urs Schreiber
(80.187.144.92)