# nLab abelian infinity-group

group theory

### Cohomology and Extensions

#### Higher algebra

higher algebra

universal 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

${\mathrm{gl}}_{1}\phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}{\mathrm{CRing}}_{\infty }\to {\mathrm{AbGrp}}_{\infty }$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)

${\mathrm{CRing}}_{\infty }\stackrel{\stackrel{𝕊\left[-\right]}{←}}{\underset{{\mathrm{gl}}_{1}}{\to }}{\mathrm{AbGrp}}_{\infty }\phantom{\rule{thinmathspace}{0ex}}.$CRing_\infty \stackrel{\overset{\mathbb{S}[-]}{\leftarrow}}{\underset{gl_1}{\to}} AbGrp_\infty \,.

This is (ABGHR 08, theorem 2.1).

## Examples

E-∞ operadE-∞ algebraabelian ∞-groupE-∞ space, if grouplike: infinite loop space $\simeq$ Γ-spaceinfinite loop space object
$\simeq$ connective spectrum$\simeq$ connective spectrum object