Universal bundles over algebraic stacks

Question

$\DeclareMathOperator\el{ell}$In the topological case, given a group $G$, we can define the classifying space $BG$ for principal $G$-bundles and there is a universal $G$-principal bundle $EG \to BG$ classified by the identity morphism $BG \to BG$ such that any other principal bundle over $X$ is given by a pullback of the universal bundle along some map $X \to BG$. Is there an analog construction for algebraic stacks?

That is, take some algebraic stack classifying a family of objects over a given scheme, e.g. the moduli stack of elliptic curves. Is there any way to make sense of "the universal elliptic curve classified by the identity morphism $\mathcal{M}_{\el} \to \mathcal{M}_{\el}$"? My problem is I'm not sure how to evaluate an algebraic stack on an algebraic stack - we can glue the value of a stack along a cover of a scheme, but is there any reason to expect a similar property for covers of algebraic stacks? Will the resulted object be an algebraic stack?

I've found this "universal bundle" mentioned in this nLab article, where they even take its bundle of differential forms, but I couldn't find any reference or explanation, so any reference would be very welcome.

Answer 1

$\DeclareMathOperator{\C}{\mathbb C}$A stack on a site $\mathcal C$ is a sheaf of $\infty$-groupoids $\mathcal C^{op}\to\mathcal S$. In particular, given two stacks $X,Y$, we can consider the $\infty$-groupoid $[X,Y]$ of natural transformations $X\Rightarrow Y$. If $X = y(C):D\mapsto \mathcal C(D,C)$ is (the sheafification of) the functor represented by $C\in\mathcal C$, the Yoneda lemma gives $[X,Y]\simeq Y(C)$, so in general you can think of $[X,Y]$ as "$Y$ evaluated on $X$".

Now if there is an representable epimorphism $y(C)\to X$ from a represented functor, we get a simplicial diagram $X_\bullet\in \mathcal C^{\Delta^{op}}$ such that $C\times_{X}C\times_{X}\dots\times_{X} C\simeq y(X_n)$ with $X\simeq \operatorname{colim}_{\Delta^{op}} y(X_n)$, and this gives $Y(X)\simeq \lim_{\Delta} Y(X_n)$. In other words, you can recover the "value of $Y$ on $X$" by its value on the cover $C$, together with descent data encoded in the higher terms of this cosimplicial diagram.

Let me work out your example of the universal elliptic curve (for convenience, over $\C$) in this language: By definition, it is the identity transformation $\mathcal M_{ell}\to \mathcal M_{ell}$. There is a representable epimorphism $\mathbb H\to\mathcal M_{ell}$: By definition, such a map is given by an elliptic curve $E_u$ over $\mathbb H$, namely $E_u = \mathbb H\times\C/\big((\tau,z)\sim (\tau,z + m + n\tau)\text{ for }m,n\in\mathbb Z\big)$. This map is representable: Given any elliptic curve $E\to Y$, the pullback $Y\times_{\mathcal M_{ell}} \mathbb H$ is the $SL(2,\mathbb Z)$-bundle over $Y$ determined by the local system of the first homology of the fibers of $E\to Y$. Thus the simplicial object is $\mathbb H\times SL(2,\mathbb Z)\times\dots\times SL(2,\mathbb Z)$, with the face maps given by the action $\tau\mapsto \frac{a\tau + b}{c\tau + d}$ of $SL(2,\mathbb Z)$ on $\mathbb H$. The universal elliptic curve is then defined by the obvious extension of this action to $E_u$ which multiplies the variable $z$ by $(c\tau +d)$, defining a "descent datum" allowing us to descend $E_u$ to $\mathbb H//SL(2,\mathbb Z)\simeq \mathcal M_{ell}$. This is the universal elliptic curve: For any base $Y$ with a principal $SL(2,\mathbb Z)$-bundle $P\to Y$, a $SL(2,\mathbb Z)$-equivariant map $P\to\mathbb H$ defines a descent datum for $P\times_{\mathbb H} E_u$, which defines an elliptic curve over $Y$, and the resulting functor is an equivalence of categories.