nLab
circle n-group

Idea

For $H$ 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 $H$. Accordingly their quotient

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

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

$$B^{n}U(1)\in H$$\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 $(n+1)$-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 $BU(1)$ can be identified with the strict 2-group whose corresponding crossed module of groups is simply $[U(1)\to 1]$.

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

Properties

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

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

Related concepts

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