nLab
Picard 3-group

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

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 structure induced on the core of the full sub-2-groupoid PIC(𝒞,)PIC(\mathcal{C}, \otimes) on those 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 September 28, 2019 at 01:54:57. See the history of this page for a list of all contributions to it.