group theory

# Contents

## Idea

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

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

under the canonical embedding $ℤ↪ℝ$ exists in $H$ and is an abelian group object: the circle group. Therefore for all $n\in ℕ$ the delooping

${B}^{n}U\left(1\right)\in H$\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 $\left(n+1\right)$-group .

## Definition

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

## Examples

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

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

## Properties

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

$\mid {B}^{n}U\left(1\right)\mid \simeq {B}^{n}U\left(1\right)\simeq {B}^{n+}{}^{ℤ}\simeq K\left(ℤ,n+1\right)\phantom{\rule{thinmathspace}{0ex}}.$|\mathbf{B}^n U(1)| \simeq B^{n} U(1) \simeq B^{n+}^\mathbb{Z} \simeq K(\mathbb{Z}, n+1) \,.

A circle $n$-group-principal ∞-bundle is a circle n-bundle, equivalently an $\left(n-1\right)$-bundle gerbe.

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