nLab
infinity-group

Context

Group Theory

(,1)(\infty,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 a grouplike A 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

(∞,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 ∞-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object

References

(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 Grpd\infty Grpd by groupoidal complete Segal spaces are discussed in

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 nrn \leq r is discussed in

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

Discussion of \infty-groups in homotopy type theory:

category: ∞-groupoid