Moduli stack of vector bundles

For the base category, let C be the category of schemes of finite type over a fixed field k. Then ${\displaystyle \operatorname {Vect} _{n}}$ is the category where

1. an object is a pair ${\displaystyle (U,E)}$ of a scheme U in C and a rank-n vector bundle E over U
2. a morphism ${\displaystyle (U,E)\to (V,F)}$ consists of ${\displaystyle f:U\to V}$ in C and a bundle-isomorphism ${\displaystyle f^{*}F{\overset {\sim }{\to }}E}$.

Let ${\displaystyle p:\operatorname {Vect} _{n}\to C}$ be the forgetful functor. Via p, ${\displaystyle \operatorname {Vect} _{n}}$ 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 ${\displaystyle \operatorname {Vect} _{n}(U)=p^{-1}(U)}$ 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).

• classifying stack
• moduli stack of principal bundles

• Behrend, Kai (2002). "Localization and Gromov-Witten Invariants". In de Bartolomeis; Dubrovin; Reina (eds.). Quantum Cohomology. Lecture Notes in Mathematics. Lecture Notes in Mathematics. Vol. 1776. Berlin: Springer. pp. 3–38.