nLab
circle n-group

Contents

Idea

For H\mathbf{H} a cohesive (∞,1)-topos such as ETop∞Grpd or Smooth∞Grpd, both the natural numbers \mathbb{Z} and the real numbers are naturally abelian group objects in H\mathbf{H}. Accordingly their quotient

U(1):=/ U(1) := \mathbb{R}/\mathbb{Z}

under the canonical embedding \mathbb{Z} \hookrightarrow \mathbb{R} exists in H\mathbf{H} and is an abelian group object: the circle group. Therefore for all nn \in \mathbb{N} the delooping

B nU(1)H \mathbf{B}^n U(1) \in \mathbf{H}

exists and has the structure of an abelian (n+1)-group object. This is the topological or smooth, respectively, circle (n+1)(n+1)-group .

Definition

Details for the smooth case are at smooth ∞-groupoid in the section circle Lie n-group .

Examples

For n=1n = 1 the circle 2-group BU(1)\mathbf{B}U(1) can be identified with the strict 2-group whose corresponding crossed module of groups is simply [U(1)1][U(1) \to 1].

Generally, for any nn B n1U(1)\mathbf{B}^{n-1}U(1) is an n-group that corresponds under the Dold-Kan correspondence to the chain complex or crossed complex of groups U(1)[n]U(1)[n] concentrated in degree nn.

Properties

The geometric realization of the circle nn-group is the Eilenberg-MacLane space

B nU(1)B nU(1)B n+K(,n+1). |\mathbf{B}^n U(1)| \simeq B^{n} U(1) \simeq B^{n+}^\mathbb{Z} \simeq K(\mathbb{Z}, n+1) \,.

A circle nn-group-principal ∞-bundle is a circle n-bundle, equivalently an (n1)(n-1)-bundle gerbe.

Revised on January 4, 2013 04:28:21 by Urs Schreiber (89.204.135.106)