$\DeclareMathOperator\el{ell}$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](https://ncatlab.org/nlab/show/modular+form#AsSections), 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.



Universal bundles over algebraic stacks
$\DeclareMathOperator{\C}{\mathbb C}$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.


