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This is a deliberately vague question, possibly obvious to experts. Let $F$ be a field. Over the (say, fpqc) site of $F$-schemes, we may define a presheaf $T^{\textrm{pre}}$ that takes a scheme $S$ and sends it to all trivial (finite rank) vector bundles over $S$. (Let me remain vague about what precisely this means.) Take the sheafification of $T^{\textrm{pre}}$. We obtain a fpqc sheaf $T$ over the site of $F$-schemes. I want $T$ to be (something like) the moduli of (finite rank) vector bundles.

Is something like this close to true? What might be the precise statement I am looking for? Any references would be appreciated.

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  • $\begingroup$ The sheafification is usually called stackification, since you get an fpqc stack, and indeed you get the stack of vector bundles that are fpqc-locally trivial. I'm not sure off the top of my head if this is the same stack as Zariski-locally trivial vector bundles (it's late here!) $\endgroup$
    – David Roberts ♦
    Commented Jan 31, 2022 at 12:48
  • $\begingroup$ But you may want to be a little careful about what you mean by the presheaf: is it valued in groupoids or categories? Both are valid, and both stackifications work fine, and will be a stack of groupoids or categories resp., but people seem think about these slightly differently. $\endgroup$
    – David Roberts ♦
    Commented Jan 31, 2022 at 12:50

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If $G$ is an affine groupe scheme over some base $S$, you can consider the groupoid $G\rightrightarrows S$. The corresponding prestack $[G\rightrightarrows S]^{pre}$ is (equivalent to) the prestack of trivial $G$-torsors. The corresponding stack $[G\rightrightarrows S]$ is (equivalent to) the stack of $G$-torsors. So it is right to think of the later as the stackification of the former. You can find details in Laumont and Moret-Bailly's book or in Olsson's book.

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    $\begingroup$ Look at the definition of a groupoid in Olsson's book: 3.4.4 p.79. $\endgroup$
    – Niels
    Commented Jan 31, 2022 at 9:33
  • $\begingroup$ The notation is meant to indicate the source and target of the groupoid scheme arising from the algebraic group. In this instance, both of those arrows are the (same) structure morphism mapping the group scheme $G$ to the base scheme $S$. In the topological category, one might take a topological group, and consider (as one can, there) $S=\ast$. Then there would be a groupoid with a single object. $\endgroup$
    – David Roberts ♦
    Commented Jan 31, 2022 at 12:46
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I'm not as familiar with the language of your setting but if the title is the spirit of what you're asking I'll answer to that...

Let $VB: Man \to sSet$ be the groupoid-valued simplicial presheaf (simplicial is probably over-powered but it's the setting I'm used to working in) which assigns to each manifold the nerve of the groupoid of bundles over $M$ and bundle-isomorphisms. Let $Prod \hookrightarrow VB$ be the sub-simplicial presheaf of product bundles. Then we can show that the sheafification of $Prod$, which assigns to each $X \in Man$ the colimit of the simplicial mapping space of simplicial presheaves $Prod^{\dagger}= colim_{U} \underline{sPre}(U, Prod)$, taken over all covers. We can check explicitly that $Prod^{\dagger}$ is object-wise weakly equivalent to $VB$ and I think this is the statement you want.

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