# nLab infinity-group

group theory

### Cohomology and Extensions

#### $(\infty,1)$-Category theory

(∞,1)-category theory

# Contents

## Definition

An ∞-group is a group object in ∞Grpd.

Equivalently (by the delooping hypothesis) it is a pointed connected $\infty$-groupoid.

Under the identification of ∞Grpd with Top this is known as an $A_\infty$-space, for instance.

An $\infty$-Lie group is accordingly a group object in ∞-Lie groupoids. And so on.

## Properties

For details see groupoid object in an (∞,1)-category.

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ Γ-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
stabilizationspectrumspectrum object

## References

The homotopy theory of $\infty$-groups that are n-connected and r-truncated for $r \leq n$ is discussed in

• A.R. Garzón, J.G. Miranda?, Serre homotopy theory in subcategories of simplicial groups Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123

category: ∞-groupoid

Revised on May 29, 2014 08:55:16 by Urs Schreiber (89.204.130.229)