# nLab Picard 2-group

Contents

### Context

#### Monoidal categories

monoidal categories

group theory

# Contents

## Definition

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

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

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