nLab Picard 2-group

Definition

The Picard groupoid $PIC(\mathcal{C}, \otimes)$ of a monoidal category $(\mathcal{C}, \otimes)$ is the core of its full subcategory on those objects that are invertible objects under the tensor product. This inherits the monoidal structure from $(\mathcal{C}, \otimes)$ and hence becomes a 2-group. This is the Picard 2-group of $(\mathcal{C}, \otimes)$.

In geometric contexts this is also called the Picard stack.

Properties

Relation to Picard group

The decategorification of the Picard 2-group, hence the group of connected components, is the ordinary Picard group $Pic(\mathcal{C}, \otimes)$.

$$Pic(\mathcal{C}, \otimes) \simeq \pi_0 PIC(\mathcal{C}, \otimes) \,.$$

Related concepts

• Picard 3-group

• Picard ∞-group

References

• Alexei Davydov, Dmitri Nikshych, Braided Picard groups and graded extensions of braided tensor categories, (arXiv:2006.08022)

Early discussion of 2-groups as Picard 2-groups:

• Alexandru Solian, Groupe dans une catégorie, C. R. Acad. Sc. Paris 275 (1972) [ark:/12148/bpt6k57310477/f7]

• Alexandru Solian, Coherence in categorical groups, Communications in Algebra 9 10 (1980) 1039-1057 [doi:10.1080/00927878108822631]