I am reading some papers concerning the moduli stack of vector bundles and there is some notion that I don't understand. Let us consider $\text{Vect}_{n}$ the stack of vector bundles with isomorphisms over the category of $k$-schemes, being $k$ an algebraically closed field of characteristic $0$. Given another stack $\mathcal{X}$ over $\text{Sch}_{k}$, some authors considered a vector bundle over $\mathcal{X}$. My question is, what is a vector bundle of rank $n$ over $\mathcal{X}$? A priori I thought that maybe a vector bundle over $\mathcal{X}$ is just a morphism of stacks $\mathcal{X}\rightarrow\text{Vect}_{n}$, but after this first impression, people start to work with a $\textit{vector bundle}$ $E\rightarrow \mathcal{X}$ as in the usual case of $k$-schemes. They even consider the pullback of a vector bundle on $\mathcal{X}$ by a morphism of stacks $\mathcal{Y}\rightarrow\mathcal{X}$. Could you help me to grasp this notion or at least give me some references?


1 Answer 1


I am trying to answer (or rather discuss) your question.

Yoneda lemma for stacks says the following:

Let $\mathcal{S}$ be a stack(differentiable, algebraic, topological,..etc). Let $\bar{\mathcal{M}}$ denote the canonical stack associated to a space $M$. Then there is a canonical equivalence of categories between $\mathcal{S}(M)$ and $\rm{Mor}_{stacks}(\bar{\mathcal{M}},\mathcal{S})$ where the later denote the category of morphism of stacks.

Now, consider the stack $\rm{Vect}_{n}$ over algebraic spaces. Now applying the above Yoneda lemma of stacks we get an equivalence of categories between $\rm{Vect}_{n}(M)$ and $\rm{Mor}_{stacks}(\bar{\mathcal{M}},\rm{Vect}_{n})$.

I think you are asking whether we can extend this result for any general stack $\mathcal{X}$ or not. (i.e not necessarily stacks of the form $\bar{\mathcal{M}}$).

PS: I am not completely aware of such "generalised form of Yoneda lemma" as of now. If I find anything in that direction I will add it in the answer. You may find something in that direction in the page 10 of SOME NOTES ON DIFFERENTIABLE STACKS by J. Heinloth https://www.uni-due.de/~hm0002/stacks.pdf


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