Intro - 2-categorical G_m reps

Let $C$ be a stable $\infty$-category, a 2-categorical rep of ${\mathbb{G}}_m$ in $C$ is equivalently data

• an autoequivalence: $T\in Au{t}_{Cat}(C)$
• a monoidal functors ${\mathbb{Z}}^{ds}\to Fun(C,C)$

By the tensoring of $Ca{t}_{St}$ over $Cat$, this corresponds to a categorified group algebra

That is, we’ll

$Ca{t}_{Gr}:=LMo{d}_{Gr}(Ca{t}_{st})$

We’re to think of $T$ as a translation/shift, and write $X(n):=T^n(X)$

Torus Action

the associativity of this shift operation is encoded by a monoidal functor

$$\mu :B\mathbb{T}\to Ca{t}_{Gr}$$

More precisely, perhaps, the map goes

${\mathbb{Z}}^{ds}\to Fun(Ca{t}_{Gr},Ca{t}_{Gr})$ $n\mapsto \left((C,T)\mapsto (C,T^n)\right)$ Exercise: Say this in terms of Picard/Brauer group.