nLab
path infinity-groupoid

**∞-Lie theory** ## 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

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Contents

Idea

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

Definition

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

Δ SmoothSp:ΔSmoothSp \Delta_{SmoothSp} : \Delta \to SmoothSp

be the canonical cosimplicial object in smooth spaces that sends the abstract nn-simplex Δ[n]\Delta[n] to the standard smooth nn-simplex Δ n n\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:SmoothSpSmoothSp Δ op N : SmoothSp \to SmoothSp^{\Delta^{op}}

with values in smooth ∞-stacks given by

N(X):USmoothSp(U×Δ SmoothSp ,X), 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):UEx SmoothSp(U×Δ SmoothSp ,X), N(X) : U \mapsto Ex^\infty SmoothSp(U \times \Delta^\bullet_{SmoothSp}, X) \,,

which can be thought of as having in degree kk piecewise smooth kk-dimensional paths.

Connections

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.

References

A more detailed account is at

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