nLab
Picard infinity-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

For (𝒞,)(\mathcal{C}, \otimes) a monoidal (∞,1)-category, its Picard \infty-group is the ∞-group induced on the full sub-∞-groupoid PIC(𝒞,)PIC(\mathcal{C}, \otimes) on the objects that are invertible under the tensor product.

Properties

Relation to Brauer \infty-group and \infty-group of units

For EE an E-∞ ring and Mod(E)Mod(E) its (∞,1)-category of ∞-modules, then the Picard \infty-group is a “non-connected delooping” of the ∞-group of units in that

ΩPic(E)GL 1(E). \Omega Pic(E)\simeq GL_1(E) \,.

Conversely Pic()Pic(-) itself has a further non-connected delooping by the Brauer ∞-group Br()Br(-) in that

ΩBr(E)Pic(E) \Omega Br(E)\simeq Pic(E)

(Szymik 11, theorem 5.7).

References

See also the discussion of higher Brauer groups in stable homotopy theory (which in turn are a “non-connective delooping”of Pic()Pic(-)) in

Last revised on May 21, 2014 at 13:12:46. See the history of this page for a list of all contributions to it.