nLab circle n-group

Idea

For $\mathbf{H}$ 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 $\mathbf{H}$. Accordingly their quotient

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

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

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

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

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

Properties

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

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

Related concepts

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