nLab Picard 2-group

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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.

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) \,.

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

Last revised on June 18, 2020 at 06:27:31. See the history of this page for a list of all contributions to it.