# nLab Picard infinity-group

Contents

### Context

#### Monoidal categories

monoidal categories

group theory

# Contents

## Definition

For $(\mathcal{C}, \otimes)$ a monoidal (∞,1)-category, its Picard $\infty$-group is the ∞-group induced on the full sub-∞-groupoid $PIC(\mathcal{C}, \otimes)$ on the objects that are invertible under the tensor product.

## Properties

### Relation to Brauer $\infty$-group and $\infty$-group of units

For $E$ an E-∞ ring and $Mod(E)$ its (∞,1)-category of ∞-modules, then the Picard $\infty$-group is a “non-connected delooping” of the ∞-group of units in that

$\Omega Pic(E)\simeq GL_1(E) \,.$

Conversely $Pic(-)$ itself has a further non-connected delooping by the Brauer ∞-group $Br(-)$ in that

$\Omega Br(E)\simeq Pic(E)$

## References

See also the discussion of higher Brauer groups in stable homotopy theory (which in turn are a “non-connective delooping”of $Pic(-)$) in

Last revised on May 21, 2014 at 13:12:46. See the history of this page for a list of all contributions to it.