With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
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.
The Picard 3-group, or rather the monoidal 2-category that it sits in, was maybe first made explicit in the last part of
The corresponding Kan complex is discussed in
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
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.
Last revised on May 26, 2023 at 19:49:31. See the history of this page for a list of all contributions to it.