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.

Proposition

Relation to commutative $\infty$-rings

Definition

Write

$$gl_1 \; \colon \; CRing_\infty \to AbGrp_\infty$$

for the (∞,1)-functor which sends a commutative ∞-ring to its ∞-group of units.

Definition

The ∞-group of units (∞,1)-functor of def. is a right-adjoint (∞,1)-functor (or at least a right adjoint on homotopy categories)

$$CRing_\infty \stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}} AbGrp_\infty \,.$$

This is (ABGHR 08, theorem 2.1).

Examples

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

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

• A 2-truncated abelian $\infty$-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 $\simeq$ ∞-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

General discussion is in section 5 of

• Jacob Lurie, Higher Algebra

Discussion in the context of E-∞ rings and twisted cohomology is in

• Matthew Ando, Andrew Blumberg, David Gepner, Michael Hopkins, Charles Rezk, Units of ring spectra and Thom spectra (arXiv:0810.4535)