nLab
infinity-group

Definition

An ∞-group is a group object in ∞Grpd.

Equivalently (by the delooping hypothesis) it is a pointed connected $\mathrm{\infty }$-groupoid.

Under the identification of ∞Grpd with Top this is known as an $A_{\mathrm{\infty }}$-space, for instance.

An $\mathrm{\infty }$-Lie group is accordingly a group object in ∞-Lie groupoids. And so on.

Properties

For details see groupoid object in an (∞,1)-category.

Examples

• Picard infinity-group

Related concepts

• group

• 2-group, braided 2-group

• n-group, k-tuply groupal n-groupoid

• $\mathrm{\infty }$-group, braided ∞-group

  • looping and delooping

  • automorphism ∞-group

    • ∞-group of bisections

  • center of an ∞-group

  • normal morphism of ∞-groups

  • stabilizer ∞-group

  • outer automorphism ∞-group

  • Eilenberg-MacLane object, ∞-gerbe

  • ∞-group extension

	(∞,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 $\mathrm{\infty }$-groups that are n-connected and r-truncated for $r\le n$ 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