nLab Picard infinity-group

Definition

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

Properties

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

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

$$\Omega Pic(E)\simeq GL_1(E) \,.$$

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

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

(Szymik 11, theorem 5.7).

Related concepts

• Picard group, Picard stack

  • Brauer group

• Picard 2-group

• Picard 3-group

• Picard $\infty$-group, Picard ∞-stack

References

• Steffen Sagave, Spectra of units for periodic ring spectra (arXiv:1111.6731)

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

• Markus Szymik, Brauer spaces for commutative rings and structured ring spectra (arXiv:1110.2956)

• Andrew Baker, Birgit Richter, Markus Szymik, Brauer groups for commutative $\mathbb{S}$-algebras, J. Pure Appl. Algebra 216 (2012) 2361–2376 (arXiv:1005.5370)