Descent over Cartier stack

Here we collect some results on $1$ and $2$- categorical descent on the stack ${\mathbb{A}}^1/{\mathbb{G}}_m$. This will allow us to explain a relationship Complex Orientation and coherent Cochains.

We have the following $B\mathbb{T}$-equivariant $(\infty ,2)$ categories

• $B\mathbb{T}\curvearrowright Ca{t}_{st}:=\{\text{small stable }\infty \text{-categories}\}$ (Waldhaus 3.5.12)
• $B\mathbb{T}\curvearrowright Ca{t}_{fil}:=\{\text{small stable }\infty \text{-categories tensored over }Fil\}$ (Waldhaus 3.5.17)

Waldhaus.pdf constructs a $B\mathbb{T}$-equivariant map of the underlying $(\infty ,1)$-categories

$Ca{t}_{st}\stackrel{MF}{\to }Ca{t}_{Fil}$ It’s a categorified version of the Sheering map, which is the functor (Waldhaus 3.5.9)

$$\Phi :Gr\to Sp$$

$X_{\bullet }\mapsto {⨁}_{n\in \mathbb{Z}}{\Sigma }^{2n}{X}_n$ Remarkably, this turns out to be a $E_2$-monoidal functor - we’ll come back to this point.

In the setting of 2-categorical G_m reps, there’s a symmetric monoidal functor $\mu :B\mathbb{T}\to Ca{t}_{Gr}$ Composing with $\Phi$ we get a $\mathbb{T}$ action on $Sp$. In words, $B\mathbb{T}$ acts on $Ca{t}_{st}$ by applying sheering to the translation symmetry of graded spectra.

Use sheering to understand Hochschild invariants!