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

Related concepts and Examples

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

• Julia Bergner,

  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≤rn \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:

• Roman Mikhailov, Homotopy patterns in group theory, Proceedings of the ICM 2022 (arXiv:2111.00737)

Discussion of

∞

\infty

-groups in homotopy type theory: