nLab Picard 3-group

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.

• If $(\mathcal{C}, \otimes)$ is a braided monoidal 2-category, then $PIC(\mathcal{C}, \otimes)$ is a braided 3-group.

• If $(\mathcal{C}, \otimes)$ is a sylleptic monoidal 2-category, then $PIC(\mathcal{C}, \otimes)$ is a sylleptic 3-group.

• If $(\mathcal{C}, \otimes)$ is a symmetric monoidal 2-category, then $PIC(\mathcal{C}, \otimes)$ is a abelian 3-group.

  In this case this is the 3-truncation of the Picard ∞-group.

Examples

• For $R$ a commutative ring and $2Mod_R$ the 2-category of 2-modules over $R$, hence of associative algebras over $R$ and bimodules between them and intertwiners between those, the corresponding Picard 3-group may be thought of as that of line 2-bundles over $Spec R$. It has as homotopy groups the Brauer group, the Picard group and the group of units of $R$.

Related concepts

• Picard group, Brauer group

• Picard 2-group

• Picard ∞-group

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

• Ross Street, Categorical and combinatorial aspects of descent theory (arXiv:math.CT/0303175)

A refinement to stable homotopy theory is discussed in

• Steffen Sagave, Spectra of units for periodic ring spectra (arXiv:1111.6731)

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

• Markus Szymik, Brauer spaces for commutative rings and structured ring spectra (arXiv:1110.2956)

• Andrew Baker, Birgit Richter, Markus Szymik, Brauer groups for commutative $\mathbb{S}$-algebras, J. Pure Appl. Algebra 216 (2012) 2361–2376 (arXiv:1005.5370)

See also:

• Urs Schreiber, Picard and Brauer 2-groups, String Theory Coffee Table, 2006.

• John Baez, The Brauer 3-group, $n$-Category Café, 2020.