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.



Universal principal bundle on stack
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.


