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 general non-abelian ∞-groups, over braided $\infty$-groups, sylleptic $\infty$-groups …, to ever more abelian groups By an abelian ∞-group (not an established term) one may want to mean an ∞-group which is maximally abelian, in this sense.

Technically, the level of abelianness may be encoded (see at May recognition theorem) by the $E_n$-operads as $n$ ranges from 1 to $\infty$: On the non-abelian end, a general ∞-group is equivalently a groupal algebra over $E_1$, also known as the associative operad, hence is a groupal A-∞ algebra; while at the abelian end a groupal $E_\infty$-space is an infinite loop space or connective spectrum. See also the periodic table of $k$-tuply monoidal $n$-groupoids.

Notice that referring to connective spectra as “abelian $\infty$-groups” (which is not standard) matches the established terminology for non-abelian cohomology (which is standard): The qualifier “non-abelian” in non-abelian cohomology is in contrast to Whitehead-generalized cohomology theories which are represented by spectra.

In a more restrictive sense one may say that plain abelian cohomology is just ordinary cohomology theory, subsuming only those Whitehead-generalized cohomology theories which are represented specifically by Eilenberg-MacLane spectra. Under the Dold-Kan correspondence these are equivalently chain complexes of abelian groups. One may think of these as being yet more commutative than general spectra and might want to reserve the term “abelian $\infty$-group” for them.

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)