nLab infinity-group

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

• model structure on simplicial groups

• model structure on reduced simplicial sets

Related concepts and Examples

• group

• 2-group, braided 2-group

• n-group, k-tuply groupal n-groupoid

• pointed connected groupoid

• $\infty$-group, braided ∞-group

  • looping and delooping

  • automorphism ∞-group

    • ∞-group of bisections

  • ∞-group of units

    • Picard ∞-group

      • Brauer ∞-group

  • p-compact group

  • center of an ∞-group

  • normal morphism of ∞-groups

  • stabilizer ∞-group

  • outer automorphism ∞-group

  • Eilenberg-MacLane object, ∞-gerbe

  • ∞-group cohomology

  • ∞-group extension

  • augmented ∞-group

  • ∞-group completion

  • finite ∞-group

• ∞-action

	(∞,1)-operad	∞-algebra	grouplike version	in Top	generally
	A-∞ operad	A-∞ algebra	∞-group	A-∞ space, e.g. loop space	loop space object
	E-k operad	E-k algebra	k-monoidal ∞-group	iterated loop space	iterated loop space object
	E-∞ operad	E-∞ algebra	abelian ∞-group	E-∞ space, if grouplike: infinite loop space $\simeq$ ∞-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

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

• Paul Goerss, Rick Jardine, chapter V of Simplicial homotopy theory chapter V.

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

• Jacob Lurie, section 6.1.2 in Higher Topos Theory

Model category presentations of group(oid) objects in $\infty Grpd$ 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

• Jacob Lurie, (∞,2)-Categories and the Goodwillie Calculus (arXiv:0905.0462)

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:

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

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

• Ulrik Buchholtz, Floris van Doorn, Egbert Rijke, Higher Groups in Homotopy Type Theory, LICS ‘18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (arXiv:1802.04315, doi:10.1145/3209108.3209150)

• Ulrik Buchholtz, Notes on higher groups and projective spaces, 2016 (pdf)

• Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: Section 4.6 of: Symmetry (2021) $[$pdf$]$

category: ∞-groupoid