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 an $A_\infty$-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

• $\infty$-group, braided ∞-group

  • looping and delooping

  • automorphism ∞-group

    • ∞-group of bisections

  • ∞-group of units

    • Picard ∞-group

      • Brauer ∞-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

	(∞,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 $\simeq$ Γ-space	infinite loop space object
				$\simeq$ connective spectrum	$\simeq$ connective spectrum object
stabilization				spectrum	spectrum object

• looping and delooping, stabilization hypothesis

References

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

category: ∞-groupoid