2-categorical G_m reps

Idea

There’s a categorification of the $\infty$-category of graded spectra. That is, a symmetric monoidal $(\infty ,2)$ category $2Gr$ and an equivalence

$En{d}_{2Gr}(1)\simeq Gr$ Since graded spectra can be identified with representations of the spectral algebraic group ${\mathbb{G}}_m$ (the flat, not smooth version), such a categorification can be viewed as a 2-categorical ${\mathbb{G}}_m$-representation.

Additionally, we have a concrete object-wise description of a 2-categorical ${\mathbb{G}}_m$ representation is a stable $\infty$-category with an internal grading.

Construction

We define

$$2Gr:=LMo{d}_{\mathbb{Z}}(Ca{t}_{st})$$

Remark (internal gradings) An object in $2Gr$ is said to be a stable $\infty$-category with an internal grading. This terminology refers to how such an object is the data of a stable category $C$ and monoidal functor $T:\mathbb{Z}\to Fun(C,C)$ The datum $T$ is said to be an internal grading. The data of the monoidal functor provides equivalences $T(0)\simeq I{d}_C$ $T(n)\simeq T(1{)}^{\circ n}$ $T(-1)\circ T(1)\simeq T(1)\circ T(-1)\simeq T(0)\simeq I{d}_C$ That is, $T$ is completely determined by the data of a endofunctor $T(1)$, which is invertible, with a inverse given by $T(-1)$.

Remark: we could more generally define an $\infty$-category with a internal grading similarly for arbitrary small (non-stable) $\infty$-categories. Our setup here uses the stability the situation to exhibit $2Gr$ as a categorification of the stable category $Gr$, which we discuss now.

Representation-theoretic Description

Using the action of $Cat$ on $Ca{t}_{st}$, we have

$2Gr=LMo{d}_{\mathbb{Z}}(Ca{t}_{st})\simeq LMo{d}_{Sp[\mathbb{Z}]}(Ca{t}_{st})$ where the subscript on the RHS denotes the categorified group algebra $Sp[\mathbb{Z}]:=Fun(\mathbb{Z},Sp{)}^{Day}$ but we also recognize this as graded spectra. In short,

$2Gr\simeq LMo{d}_{Gr}(Ca{t}_{st})$ Geometric interpretation

There’s a symmetric monoidal equivalence arising from descent along $pt\to B{\mathbb{G}}_m$

$$Gr\simeq QCoh(B{\mathbb{G}}_m):=Rep({\mathbb{G}}_m)$$

Monadic comparison for base change of 2-categorical sheaves along $B{\mathbb{G}}_m\to pt$ (i.e. the 2-categorical global sections adjunction) gives a map $2QCoh(B{\mathbb{G}}_m)\to LMo{d}_{QCoh(B{\mathbb{G}}_m)}(Cat)\simeq 2Gr$ which would be an equivalence if $B{\mathbb{G}}_m$ is 1-affine.