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$\DeclareMathOperator\QCoh{QCoh} \newcommand\numC{\mathbb{C}} \newcommand\numQ{\mathbb{Q}}$Let $X$ be a topological space (possibly a scheme). What happens if you apply $\QCoh$ to the prestack $(X,\underline{\numC}_{X})$, where $\underline{\numC}_{X}$ is the constant sheaf at $X$? What about $(X, \underline{\numQ}_{\ell})$?

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  • $\begingroup$ You get locally constant sheaves of vector spaces. $\endgroup$ Jun 15 at 17:38
  • $\begingroup$ @MarcHoyois I thought this, but I'm not sure this is true, since you could have a locally constant sheaf on an affine variety which is not constant and thus not quasicoherent in this definition. $\endgroup$
    – Will Sawin
    Jun 15 at 19:11
  • $\begingroup$ @WillSawin That was also the answer I was hoping for, or possibly something related (perhaps providing a different approach to constructible sheaves). Unfortunately, a map $\operatorname{Spec}(A) \to (X,\underline{\mathbb{C}}_{X})$ is just any continuous map (seemingly without restrictions) and a map of sheaves of rings $\underline{\mathbb{C}}_{\operatorname{Spec}(A)} \to \mathcal{O}_{\operatorname{Spec}(A)}$. There can potentially be many such; if $A$ is a $\mathbb{C}$-algebra, any pathological automorphism of $\mathbb{C}$ provides you with one. $\endgroup$
    – Gaussler
    Jun 16 at 11:49
  • $\begingroup$ @WillSawin To clarify I was using the following definition, which I think is standard although rarely used: a sheaf of modules over a ringed space $(X,\mathcal O_X)$ is quasi-coherent if locally it is the cokernel of a map of free $\mathcal O_X$-modules (this is manifestly a local notion). For $\mathcal O_X=\underline{\mathbb C}_X$ we therefore get locally constant sheaves, at least if $X$ is locally connected or if we restrict to modules of finite presentation. $\endgroup$ Jun 16 at 12:32
  • $\begingroup$ @MarcHoyois Sorry, I confused myself about what happens when you pull back a locally constant sheaf to an affine variety. But I don't think your definition makes sense for prestacks, so I see a map from the category of Zariski-locally constant sheaves to this category of quasicoherent sheaves but not yet vice versa. $\endgroup$
    – Will Sawin
    Jun 16 at 12:43

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