nLab
Picard 2-group

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Group Theory

Contents

Definition

The Picard groupoid PIC(𝒞,)PIC(\mathcal{C}, \otimes) of a monoidal category (𝒞,)(\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 (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 September 28, 2019 at 01:47:32. See the history of this page for a list of all contributions to it.