Relative motivic stack-function groups for quiver representation stacks

Fix a field $k$. An Artin $k$-stack is as in Algebraic stacks and their inertia stacks. In this item, a finite-type stack means an Artin $k$-stack with a finite-type smooth atlas and finite-type diagonal. It has affine stabilizers when the automorphism group scheme of every geometric point is affine.

For a finite-type Artin stack $X$ with affine stabilizers, define $\mathcal{K}(X)$ to be the abelian group generated by isomorphism classes $[Y\to X]$ of finite-type Artin stacks $Y$ with affine stabilizers and a 1-morphism to $X$. Impose the following relations.

1. If $Z\hookrightarrow Y$ is a closed substack, then $[Y\to X]=[Z\to X]+[Y\setminus Z\to X]$.
2. Let $S\to X$ be a finite-type stack with affine stabilizers and let $d:V_0\to V_1$ be a constant-rank morphism of vector bundles on $S$. The additive vector bundle $V_0$ acts on $V_1$ by translation through $d$. Write $[V_1\mathrm{/}{V}_0\to S\to X]$ for the resulting quotient groupoid stack over $X$. If $r_i$ is the rank of $V_i$, impose $[V_1\mathrm{/}{V}_0\to X]={\mathbb{L}}^{r_1-r_0}[S\to X]$.

Here $\mathbb{L}$ is the class of ${\mathbb{A}}_k^1$ in $R:=\mathcal{K}(\mathrm{Spec}k)$, and we localize $R$ and the modules $\mathcal{K}(X)$ by making $\mathbb{L}$ invertible. The product in $R$ is induced by product over $k$. The group $\mathcal{K}(X)$ is an $R$-module. This explicit vector-bundle-stack relation records the class of a family of linear objects modulo its linear changes of splitting.

For a morphism $f:X\to X^{\mathrm{\prime }}$, pullback is $f^{\ast }[Y\to X^{\mathrm{\prime }}]=[Y{\times }_{X^{\mathrm{\prime }}}X\to X]$ and pushforward is $f_{!}[Z\to X]=[Z\to X\to X^{\mathrm{\prime }}]$. Exterior product is $[Y\to X]⊠[Z\to X^{\mathrm{\prime }}]=[Y{\times }_{k}Z\to X{\times }_k{X}^{\mathrm{\prime }}]$. These operations are well-defined because fiber product and composition preserve isomorphisms, closed-substack decompositions, and the vector-bundle-stack relation. If the rank of a vector-bundle map varies, its base is partitioned into the finitely many locally closed rank strata and the scissor relation is applied on those strata.

The dependency on The Axiom of Choice is inherited from the cited algebraic-stack convention in Algebraic stacks and their inertia stacks, specifically its represented-sheaf descent foundation. No choice is used in the finite-dimensional quiver matrix calculations below.