nLab
abelian infinity-group

Context

Group Theory

Higher algebra

Contents

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:CRing AbGrp 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. 1 is a right-adjoint (∞,1)-functor (or at least a right adjoint on homotopy categories)

CRing gl 1𝕊[]AbGrp . CRing_\infty \stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}} AbGrp_\infty \,.

This is (ABGHR 08, theorem 2.1).

Examples

(∞,1)-operad∞-algebragrouplike versionin Topgenerally
A-∞ operadA-∞ algebra∞-groupA-∞ space, e.g. loop spaceloop space object
E-k operadE-k algebrak-monoidal ∞-groupiterated loop spaceiterated loop space object
E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space \simeq Γ-spaceinfinite loop space object
\simeq connective spectrum\simeq connective spectrum object
stabilizationspectrumspectrum object

References

General discussion is in section 5 of

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

Revised on June 9, 2013 19:31:39 by Urs Schreiber (173.212.86.110)