nLab Picard 3-group



Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory

Group Theory



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.



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 June 18, 2020 at 02:26:37. See the history of this page for a list of all contributions to it.