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In algebraic geometry, given a smooth projective curve X over a finite field F q {\displaystyle \mathbf {F} _{q}} and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by Bun G {\displaystyle \operatorname {Bun} _{G}} , is an algebraic stack given by: for any F q {\displaystyle \mathbf {F} _{q}} -algebra R,
In particular, the category of F q {\displaystyle \mathbf {F} _{q}} -points of Bun G {\displaystyle \operatorname {Bun} _{G}} , that is, Bun G {\displaystyle \operatorname {Bun} _{G}} , is the category of G-bundles over X.
Similarly, Bun G {\displaystyle \operatorname {Bun} _{G}} can also be defined when the curve X is over the field of complex numbers. Roughly, in the complex case, one can define Bun G {\displaystyle \operatorname {Bun} _{G}} as the quotient stack of the space of holomorphic connections on X by the gauge group. Replacing the quotient stack by a homotopy quotient gives the homotopy type of Bun G {\displaystyle \operatorname {Bun} _{G}}.
In the finite field case, it is not common to define the homotopy type of Bun G {\displaystyle \operatorname {Bun} _{G}}. But one can still define a cohomology and homology of Bun G {\displaystyle \operatorname {Bun} _{G}}.