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For a cohesive (∞,1)-topos such as ETop∞Grpd or Smooth∞Grpd, both the natural numbers and the real numbers are naturally abelian group objects in . Accordingly their quotient
under the canonical embedding exists in and is an abelian group object: the circle group. Therefore for all the delooping
exists and has the structure of an abelian (n+1)-group object. This is the topological or smooth, respectively, circle -group .
Details for the smooth case are at smooth ∞-groupoid in the section circle Lie n-group.
For the circle 2-group can be identified with the strict 2-group whose corresponding crossed module of groups is simply .
Generally, for any is an n-group that corresponds under the Dold-Kan correspondence to the chain complex or crossed complex of groups concentrated in degree .
The geometric realization of the circle -group is the Eilenberg-MacLane space
A circle -group-principal ∞-bundle is a circle n-bundle, equivalently an -bundle gerbe.
Last revised on November 21, 2019 at 18:32:13. See the history of this page for a list of all contributions to it.