nLab
Picard 3-group

Contents

Context

Monoidal categories

Group Theory

Contents

Definition

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

Examples

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()Pic(-)) in

Last revised on April 1, 2014 at 10:28:06. See the history of this page for a list of all contributions to it.