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In what follows, assume several universes for simplicity.

Let $X$ be a stack in groupoids on the fpqc site of small affine schemes $\mathbf{Aff}_{\text{fpqc}}$. We can define $\mathbf{QCoh}(X)$ formally by considering the representable stack $$\mathbf{Hom}_{\operatorname{Stk}_{\text{fpqc}}(\mathbf{Aff})}(-,\mathbf{QCoh})$$ and restricting it to the full 2-subcategory $\operatorname{Stk}^{\text{Gpd}}_{\text{fpqc}}(\mathbf{Aff})^{\text{op}}$.

We can define the fpqc topos of $X$ to be the slice category $\operatorname{Shv}_{\text{fpqc}}(\mathbf{Aff})/X$, and we can make this topos locally ringed by pulling back $\mathbb{A}^1_X=X\times_{\operatorname{Spec}(\mathbb{Z})}\mathbb{A}^1.$

Since $\mathbb{A}^1_X$ is a local ring object in the topos, we can then define a category of $\mathbb{A}^1_X$-modules in $\operatorname{Shv}_{\text{fpqc}}(\mathbf{Aff})/X$. By the definition of $\mathbf{QCoh}(X)$, we can write down a forgetful functor $U$ to the category of $\mathbb{A}^1_X$-modules by sending a quasicoherent sheaf $F$ on $X$ to the evaluation of its pullback, that is, for $f:\operatorname{Spec}(R)\to X$, we have $$U(F)(f)=\Gamma(\operatorname{Spec}(R),f^\ast F),$$ which is naturally an $\mathbb{A}^1_X$-module.

Question: Is the forgetful functor $U$ defined above fully faithful? If so, can we identify its essential image as the full subcategory of $\mathbb{A}^1_X$-modules admitting a presentation the way we can for schemes? Does either of these statements become true or stay true if we pass to derived everything? What if we restrict to the case where $X$ is an Artin stack?

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  • $\begingroup$ Are both the words meant to be 'pseudofunctor' in "Kan extending the pseudofunctor … to a pseudofunctor"? $\endgroup$
    – LSpice
    Commented May 9, 2020 at 1:08
  • $\begingroup$ @LSpice Yeah, both are pseudofunctors. I guess one can do this more simply by considering the stack of categories $\mathbf{QCoh}$, taking the representable pseudofunctor $\operatorname{Hom}(-,\mathbf{QCoh})$ and restricting it to the full subcategory of groupoidal stacks, no Kan extension needed. $\endgroup$
    – Steve
    Commented May 9, 2020 at 1:18

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