nLab infinity-group



Group Theory

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



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.


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 ∞-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 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 nrn \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 (


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

Discussion of \infty -groups in homotopy type theory:

category: ∞-groupoid

Last revised on January 24, 2023 at 17:00:02. See the history of this page for a list of all contributions to it.