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I am studying the notes of Sorger concerning the moduli problem of principal bundles over curve https://inis.iaea.org/collection/NCLCollectionStore/_Public/38/005/38005695.pdf and there is something I don't quite understand. He considers the moduli stack of principal $G$-bundles over a curve $G$, and it is denoted by $\mathcal{M}_{G,C}$, being $\mathcal{M}_{G,C}(S)$ the groupoid of principal $G$-bundles over $C\times S$ for each $\mathbb{C}$-scheme $S$. In Proposition 3.6.8, he considers "the universal principal $G$-bundle on $C\times\mathcal{M}_{G,C}$" and I don't have a clue about what this is. I know that giving a principal bundle over a stack is to give for each $\mathbb{C}$-scheme $S$ a principal bundle over $S$, for each morphism of schemes $S\rightarrow S'$ an isomorphism between the principal bundle over $S$ and the pullback of the principal bundle over $S'$ by the map $S\rightarrow S'$, and in such way that the cocycle compatibility condition holds. I would really like to know what is the universal principal bundle in this case. Thank you for your time and wisdom in advance.

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    $\begingroup$ Yoneda Lemma $\endgroup$
    – abx
    Jul 5 at 13:09
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    $\begingroup$ Let $P_{univ}$ denote this universal bundle. I believe the correct definition of $P_{univ}$ is this: It is the fibered category whose objects over a scheme $S$ are pairs $(P,s)$ where $P$ is a principal $G$-bundle over $C\times S$, and $s$ is a section of $P$, and morphisms are morphisms of bundles preserving the section. The forgetful map to $M_{G,C}$ is visibly representable, and one checks that the pullback of $P_{univ}$ via any map $f : S\rightarrow M_{G,C}$ is precisely the bundle corresponding to $f$. This is probably what Sorger means. $\endgroup$
    – Will Chen
    Jul 6 at 6:37
  • $\begingroup$ Well, I don't think so @WillChen. To give a principal bundle over a stack is not about given a fibered category, but some data over each smooth atlas of your stack. I think that the correct definition is the one given by KevinVentullo below. $\endgroup$ Jul 6 at 7:23
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    $\begingroup$ @SamanthaSmith Stacks are fibered categories. Principal bundles over schemes are schemes. Principal bundles over stacks are stacks. Yes principal bundles can also be given on an atlas, but that data also defines an honest stack over $M_{G,C}$, which you might call the "total space" of the principal bundle. $\endgroup$
    – Will Chen
    Jul 6 at 16:48
  • $\begingroup$ @WillChen Oh thank you for your insight. I haven´t thought about principal bundle over stacks in that way and you are completely right. $\endgroup$ Jul 6 at 18:41
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Expanding on abx’s comment, let’s suppose $\mathcal{M}_{G,C}$ was representable as a scheme. Then for any $\mathbb{C}$-scheme $S$ there is a canonical isomorphism

$Hom_\mathbb{C}(S, \mathcal{M}_{G,C}) \cong \{\text{Groupoid of principal }G\text{-bundles over }C \times S\}$.

Now take $S = \mathcal{M}_{G,C}$, and consider the element of the LHS given by the identity map on $\mathcal{M}_{G,C}$. The corresponding element of the RHS is precisely the “the universal principal $G$-bundle on $C\times\mathcal{M}_{G,C}$". It is called this because any other principal $G$-bundle on some $C \times T$ can be uniquely realized as a pullback of this one via some morphism $T\rightarrow \mathcal{M}_{G,C}$.

Now in reality, $\mathcal{M}_{G,C}$ is not a scheme but merely an algebraic stack. However, there is still a notion of morphisms $T\rightarrow \mathcal{M}_{G,C}$ for $\mathbb{C}$-schemes $T$, and some stacky notion of bundle on it which pulls back to honest bundles on schemes. I can’t fill in the details but I hope this gives some intuition.

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    $\begingroup$ Well, after thinking about it, the use of the Yoneda lemma is compulsory. We need to consider the identity map from $\mathcal{M}_{G,C}\rightarrow\mathcal{M}_{G,C}$ and then study the equivalence between the data of a principal bundle over a stack and the morphisms of your stack of interest into $\mathcal{M}_{G,C}$. Thank you for your time. $\endgroup$ Jul 6 at 7:28

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