# nLab infinity-group

Contents

group theory

### Cohomology and Extensions

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# 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 a grouplike $A_\infty$-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.

## Models

By

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 $\simeq$ ∞-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object
stabilizationspectrumspectrum object

(For more see also the references at infinity-action.)

A standard textbook reference on $\infty$-groups in the classical model structure on simplicial sets is

Group objects in (infinity,1)-categories are the topic of

Model category presentations of group(oid) objects in $\infty Grpd$ by groupoidal complete Segal spaces are discussed in

• Adding inverses to diagrams encoding algebraic structures, Homology, Homotopy and Applications 10 (2008), no. 2, 149–174. (arXiv:0610291)

Adding inverses to diagrams II: Invertible homotopy theories are spaces, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (web, arXiv:0710.2254)

Discussion from the point of view of category objects in an (∞,1)-category is in

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

/S0022-4049(98)00143-1“>doi:10.1016/S0022-4049(98)00143-1</a>)

Discussion of aspects of ordinary group theory in relation to $\infty$-group theory:

Discussion of $\infty$-groups in homotopy type theory:

category: ∞-groupoid