# nLab Picard 3-group

Contents

### Context

#### Monoidal categories

monoidal categories

group theory

# Contents

## Definition

For $(\mathcal{C}, \otimes)$ a monoidal 2-category, its Picard 3-group or Picard-Brauer 3-group is the 3-group structure induced on the core of the full sub-2-groupoid $PIC(\mathcal{C}, \otimes)$ on those objects that are invertible under the tensor product.

## References

The Picard 3-group, or rather the monoidal 2-category that it sits in, was maybe first made explicit in the last part of

• R. Gordon, A.J. Power, Ross Street, Coherence for tricategories, Memoirs of the American Math. Society 117 (1995) Number 558.

The corresponding Kan complex is discussed in

• John Duskin, The Azumaya complex of a commutative ring, Categorical algebra and its applications (Louvain-La-Neuve, 1987), 107-117, Lecture Notes in Math., 1348, Springer, Berlin, 1988.

A summary of these considerations is in section 12 of

A refinement to stable homotopy theory is discussed in

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

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