The Picard groupoid$PIC(\mathcal{C}, \otimes)$ of a monoidal category$(\mathcal{C}, \otimes)$ is its full subcategory on the objects that are invertible objects under the tensor operation. 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$Pic(\mathcal{C}, \otimes)$.