nLab
Picard 2-group

Context

Monoidal categories

Group Theory

Contents

Definition

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

Pic(𝒞,)π 0PIC(𝒞,). Pic(\mathcal{C}, \otimes) \simeq \pi_0 PIC(\mathcal{C}, \otimes) \,.

Last revised on May 22, 2017 at 16:05:44. See the history of this page for a list of all contributions to it.