nLab
abelian infinity-group

Idea

An ordinary group is either an abelian group or not. For an ∞-group there is an infinite tower of notions ranging from completely non-abelian to completely abelian. An abelian ∞-group is one which is maximally abelian. This is equivalently a connective spectrum object.

Examples

• A 0-truncated abelian $\mathrm{\infty }$-group is equivalently an abelian group.

• A 1-truncated abelian $\mathrm{\infty }$-group is equivalently a symmetric 2-group.

• A 2-truncated abelian $\mathrm{\infty }$-group is equivalently a symmetric 3-group.

Related concepts

	(∞,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

• ∞-group, k-tuply groupal n-groupoid

• braided ∞-group,

  • braided 2-group

  • braided 3-group

• sylleptic ∞-group

  • sylleptic 3-group

• abelian ∞-group

  • abelian group

  • abelian 2-group

  • abelian 3-group

References

Section 5 of

• Jacob Lurie, Higher Algebra