Basic Objects in Spherical Derived Geometry

This collects some basic objects appearing in Setting - Spherical Derived Geometry

Brauer Stacks

• ${\mathbb{G}}_m:=Spec(\mathbb{S}[\mathbb{Z}])$, (flat) multiplicative group. It classifies “strict units” : ${\mathbb{G}}_m(R)\simeq Ma{p}_{Gp(S)}(\mathbb{Z},{\mathfrak{gl}}_1(R)))$ Via categorification, the space ${\mathfrak{gl}}_1(R):={\Omega }^{\infty }{R}^{\times }$ admit a tower of non-connective deloopings, see omega-monoids

(Remark: these are plain functors, not tensor functors… )

• the strict Picard stack $Pi{c}_{str}(R):=Ma{p}_{Gp(S)}(\mathbb{Z},Pic(R))$ Is represented by $Spec(Pi{c}^L(\mathbb{Z}))$