Moduli stack of vector bundles

In algebraic geometry, the moduli stack of rank-n vector bundles Vectn is the stack parametrizing vector bundles (or locally free sheaves) of rank n over some reasonable spaces.

It is a smooth algebraic stack of the negative dimension $$-n^2$$. Moreover, viewing a rank-n vector bundle as a principal $$GL_n$$-bundle, Vectn is isomorphic to the classifying stack $$BGL_n = [\text{pt}/GL_n].$$

Definition
For the base category, let C be the category of schemes of finite type over a fixed field k. Then $$\operatorname{Vect}_n$$ is the category where
 * 1) an object is a pair $$(U, E)$$ of a scheme U in C and a rank-n vector bundle E over U
 * 2) a morphism $$(U, E) \to (V, F)$$ consists of $$f: U \to V$$ in C and a bundle-isomorphism $$f^* F \overset{\sim}\to E$$.

Let $$p: \operatorname{Vect}_n \to C$$ be the forgetful functor. Via p, $$\operatorname{Vect}_n$$ is a prestack over C. That it is a stack over C is precisely the statement "vector bundles have the descent property". Note that each fiber $$\operatorname{Vect}_n(U) = p^{-1}(U)$$ over U is the category of rank-n vector bundles over U where every morphism is an isomorphism (i.e., each fiber of p is a groupoid).