Filtrations as generating functions

Flat ${\mathbb{A}}^1:=\mathbb{S}[\mathbb{N}]$ is flat over $\mathbb{S}$, which gives a calculation of its homotopy groups ${\pi }_{\bullet }(\mathbb{S}(t))\simeq ({\pi }_{\bullet }\mathbb{S})(t)$ Hence, functions on flat ${\mathbb{A}}^1$ are generating functions for stable homotopy groups. An homogenous element in degree 0 is a polynomial

$\alpha (t)=a_0+a_{1}t+a_2{t}^2+\cdots$ This is data of a map of plain spectra $\mathbb{S}[t]\to \mathbb{S}$, which is adjoint to a map of $\infty$-groupoids $\mathbb{N}\to {\Omega }^{\infty }\mathbb{S}$  That is, it’s the data of a virtual finite set $a_i$ for each $i\in \mathbb{N}$. Hence, we see we’re encoding generating functions for homotopy-sensitive combinatorics using virtual finite sets.

The operation sending $t\to 0$ is implemented by a ring map, which is $\mathbb{S}[-]$ applied to the final map in $CMon$ going $\mathbb{N}\to \ast$. Choosing another ring map $\mathbb{S}[t]\to \mathbb{S}$ That is, a point on ${\mathbb{A}}^1$, or a “character”, we get data which are homotopy-coherent instructions about how to substitute $t$ with another formula.

Taking into account the ${\mathbb{G}}_m$ equivariance, functions on ${\mathbb{A}}^1/{\mathbb{G}}_m$ are scale-invariant generating functions on $\mathbb{S}\simeq \mathcal{O}(pt)$. This formulation suggests a categorification of this principle:

Sheaves on ${\mathbb{A}}^1/{\mathbb{G}}_m$ are scale-invariant generating functions on $Sp\simeq QCoh(pt)$.

To make this precise,